Generalization of Curl to higher dimensions

Let ${\bf K}$ be a vector field in the neighbourhood of ${\bf p}\in{\mathbb R}^n$, and let ${\bf X}$ and ${\bf Y}$ be two tangent vectors at ${\bf p}$. These two vectors span a parallelogram $P$ with one vertex at ${\bf p}$. The "circulation" of ${\bf K}$ around $P$ computes to $$ \int_{\partial P}{\bf K}\cdot \mathrm{d}{\bf x}= (L\,{\bf X})\cdot{\bf Y}- (L\,{\bf Y})\cdot{\bf X} + o(|P|^2) $$ with $L:=\mathrm{d}{\bf K}({\bf p})$ and $|P|:= \mathrm{diam}(P)$. It follows that there is a certain skew bilinear function ${\rm Rot}{\bf K}({\bf p}):T_{\bf p}\times T_{\bf p}\to{\mathbb R}$ with $$ \int_{\partial P}{\bf K}\cdot \mathrm{d}{\bf x}={\rm Rot}{\bf K}({\bf p})({\bf X},{\bf Y})+ o(|P|^2) \quad (|P|\to 0). $$ In the case $n=3$ the bilinear form ${\rm Rot}$ can be represented by the vector ${\rm curl}{\bf K}$ in the form $$ {\rm Rot}{\bf K}({\bf p})({\bf X},{\bf Y}) = {\rm curl}{\bf K}({\bf p})\cdot({\bf X}\times{\bf Y}). $$

The curl of a vector field $X=P\partial_x+Q\partial_y+R\partial_z$ is equal to
$$ \mathrm{Curl}(X)= (R_y-Q_z)\,\partial_x +(P_z-R_x)\,\partial_y+ (Q_x-P_y)\,\partial_z $$

For the moment we replace $\partial_x,\partial_y,\partial_z$ with $dy\wedge dz,\ dx\wedge dz$ and $dx\wedge dy$ respectively (In fact we apply the Hodge star operator to the dual of the basis $\partial_x,\partial_y,\partial_z$).

So actually the component of $\mathrm{Curl}(X)$ is identical to the components of the $2$ form $$ \alpha=(R_y-Q_z)\,dy\wedge dz +(P_z-R_x)\,dx\wedge dz +(Q_x-P_y)\,dx\wedge dy $$

On the other hand the vector field $X$, being a section of the tangent bundle $T\mathbb{R}^3$, can be considered as a map
\begin{align} X\colon \mathbb{R}^3& \to \mathbb{R}^3\times \mathbb{R}^3\\ (x,y,z)&\mapsto (x,y,z, P(x,y,z),Q(x,y,z),R(x,y,z)). \end{align} Note that the following equality holds: $$ \alpha =X^* \omega, $$
where $\omega$ is the natural symplectic structure of $\mathbb{R}^3 \times \mathbb{R}^3$ with $\omega= dx\wedge dp +dy\wedge dq+dz \wedge dr$.

The situation described above is a motivation to consider the following generalization of the concept of the curl of a vector field on an arbitrary Riemannian manifold.

A generalized curl: Let $(M,g)$ be a Riemannian manifold. The metric $g$ gives an isomorphism (hence diffeomorphism) between $TM$ and $T^* M$. So the standard intrinsic symplectic structure of the cotangent bundle is carried to a symplectic structure $\omega$ on $TM$. Now assume that $X:M \to TM$ is a vector field. We define the curl of $X$ as a $2$-form with the following formula: $$ \mathrm{Curl}(X):=X^* \omega. $$

This was already mentioned at the MO question A generalization of Gradient vector fields and Curl of vector fields.