Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?

This is rather standard, though, unfortunately this knowledge is confined to a rather narrow segment of mathematicians working on PDEs. The main reason is that, most of the time, this is not necessary for practical problems. In fact there are multiple ways converting a PDE into invariant form.

In mathematical physics, this question comes up when trying to generalized equations from flat space to curved space. The main technical device is to introduce explicit dependence on a metric $g$ and use covariant differentiation, with respect to $g$, to replace ordinary derivatives. The invariant form of the original flat space equations is then obtained by simply replacing $g$ by the Euclidean metric. This method is sometimes colloquially known as the comma goes to semicolon rule, where the punctuation signs represent notation for ordinary and covariant derivatives. More on this can be found in textbooks on GR. For instance, Gravitation by Misner, Thorne & Wheeler would definitely discuss this.

I would guess that you might be happy to stop here. But if you are looking for more generality and mathematical sophistication, read on.

Another method is to repeatedly introduce auxiliary dependent variables, until the PDE system becomes first order and this can be done in such a way that all dependent variables are actually differential forms and differentiations are only effected through the exterior derivative. Once in this form, the PDE system is manifestly invariant. This approach was pioneered by Cartan and is explained in depth in the well known book Exterior Differential Systems by Bryant et al.

Yet another way, which does not involve reduction to a first order system, is to notice that derivatives of arbitrary order are mathematically described in an invariant way by jets. An arbitrary PDE can then be described in terms of jets. Once one unwinds all the relevant definitions, this kind of description is manifestly invariant, but not necessarily always helpful, since all it really does is bundle up the complicated rules for transforming higher derivatives under coordinate changes into the language of jets. Nevertheless, I think it is important to learn about it, as it provides the tools to study PDE systems in a very deep way. An elementary treatment of PDEs in the context of jets can be found in the book Applications of Lie Groups to Differential Equations by Olver. A much more sophisticated treatment can also be found in the aforementioned book by Bryant et al. Another very helpful reference is Natural Operations in Differential Geometry by Kolar, Michor & Slovak.

Let $\mathcal M$ be a smooth manifold. A linear PDE on $\mathcal M$ is a sum of terms $$ X_1\dots X_N u,\quad\text{where the $X_j$ are smooth vector fields.} $$ We may use the convention that if $N=0$, this is just the multliplication by a smooth function. In other words a linear PDE on $\mathcal M$ is a (non-commutative) polynomial with constant coefficients of vector fields. This PDE is said to be of order $m$ when the polynomial is of degree $m$. A general PDE on $\mathcal M$ is a function $F(u,P_1 u,\dots, P_m u)$ where $P_j$ is a linear differential operator of order $j$ defined above.

Let me give a simple celebrated example. Euler system describing non-viscous incompressible fluids is usually written as $$ \partial_t u+u\cdot \nabla u=-\nabla p,\quad \text{div}u=0,\quad u_{\vert t=0}=u_0. $$ The function $u$ is defined on $\mathbb R_t\times\mathbb R^3_x$ and valued in $\mathbb R^3_x$. A geometric version on a $3D$ manifold is shedding light on this system of equations. In the first place $u$ should be considered as a 1-form, the pressure $p$ is a function and $dp$ is its differential, also a 1-form. Now suppose that we have on our manifold an additional structure, such as a Riemannian structure allowing to identify 1-forms to vector fields. In the case of a Riemannian structure given by a metric tensor $g$, we may identify a 1-from $u$ with a vector field $v$ through the pairing $$ g(v,w)=\langle u,w\rangle,\quad \text{for any vector field $w$. We write this as $gv=u$}. $$ We may then consider the Lie derivative $\mathcal L_v$ and in coordinates in $\mathbb R^3$ $$ \sum_{1\le j\le 3}\mathcal L_v(u_j) dx_j=\mathcal L_v(u)-\sum_{1\le j\le 3}u_j \mathcal L_v(dx_j)=\mathcal L_v(u)-\sum_{1\le j\le 3}u_j dv_j=\mathcal L_v(u)-\frac12 d(\vert v\vert^2). $$ Euler system expresses the exactness of a 1-form and reads in a simply connected 3D Riemannian manifold $$ \partial_t u+\mathcal L_v(u)-\frac12 d(g(v,v))=-dp, gv=u, \text{div $v$=0} $$ so that (note that $\mathrm{curl}\, u$ appears simply as the exterior differential $du$: two-forms in 3D are also three dimensional), since the Lie derivative commutes with the exterior differentiation, with $\omega =du$, $$ \partial_t \omega +\mathcal L_v(\omega)=0,\quad gv=u,\omega =du,\quad \text{div $v$=0}. $$ This last expression provides a geometric view on Euler's system. Similar things could be done for the Navier-Stokes system.

To write the equations in invariant form you first need to define the various objects involved in invariant form. Sometime this is not so obvious and new insight is needed. Here are two examples that hopefully will illustrate ways in which things can get tricky.

1.The Euler-Lagrange equations are coordinate independent, but if the Lagrangian is a bit more sophisticated it is difficult to come-up with an invariant form. One convenient way to do this is via the Legendre transform leading to Hamiltonian equations. However, this can be used only for certain classes of Lagrangians.

2.The equations arising in gauge theory require a bit of finesse since they necessitate the introduction of a new object (e.g. a principal bundle) whose topological type hides some subtle physical quantity. The difficulties do not stop here and this topic is still being investigated from a rather sophisticated point of view. Andy Manion post, Gauge theory and the variational bicomplex is a good place to start.