How to introduce stress tensor on manifolds?

It is important to distinguish between covariant and contravariant indices of a tensor. Differential forms are totally antisymmetric covariant tensor fields. So a 2-form has 2 covariant indices, and when you swap them, the sign changes. Contravariant indices are written as upper indices and covariant indices as lower indices. You can raise and lower indices by use of a metric. Now, the stress tensor has one covariant index and one contravariant index. When you lower the contravariant index, you get a symmetric tensor field, not a differential form. In local coordinates, you simply have a matrix associated to every point, say ${\bf P}(\vec x)$.

The easiest way to understand what the stress tensor does is to imagine the effect of infinitesimal deformations inside the body, described by a vector field, say $\vec v(\vec x)$. The actual displacement at $\vec x$ could be written as $\vec v(\vec x) dr$. Now, the Energy density released by this displacement is $dE = P^j_iv^i_{;j}\ dr$, or, if you take $\vec v(\vec x)$ as a velocity, $P^j_iv^i_{;j}$ will simply be the power density. The semicolon indicates the covariant derivative. You can compute it by taking local coordiantes such that at $\vec x$ the metric is the Euclidean metric and all derivates of the metric are zero. In such local coordinates, $P^j_iv^i_{;j} = {\rm tr}({\bf PJ}_{\vec v})$, with ${\bf J}_{\vec v}$ the Jacobi matrix of $\vec v$.

Edit: What I am saying is that you cannot use differential forms alone. They are special tensors, but you need more general tensors. The stress tensor is a vector-valued 1-form (which, in 3 dimensions, is equivalent to a vector-valued 2-form, by Hodge duality, which gives a little more weight to the surface interpretation you formulated above). A vector is a contravariant 1-tensor, a 1-form is a covariant 1-tensor. Using the metric, you can transform one into the other, so you could even write the stress tensor as a 1-form-valued 1-form (or $(n-1)$-form, in $n$ dimensions), but that seems not very physical to me.

I found a paper supporting my comment that $P$ is a 1-form valued 2-form, that surface force f is also a 1-form valued 2-form, and power density is the 2-form that results from contracting f with the surface velocity. The paper is

R. Segev and L. Falach. Velocities, stresses and vector bundle valued chains. J. Elast. 105:187-206, 2011.