Let $V$ be a vector space and $\tilde{V}$ be its dual space. Let $T^{a_1 ... a_n}_{\ \ \ \ \ \ \ \ \ \ \ \ b_1 ... b_m}$ be a type $(n,m)$ tensor. By definition, it is a multlinear map
$$T: \tilde{V^n} \times V^m \to\mathbb{R}.$$
It seems like we frequently trade off on two different ways of looking at a tensor. A vector, $v^a$ is a type $(1,0)$ tensor. That is, it is (multi)linear map $v^a ( w_a ) = r \in \mathbb{R}.$
So we have the geometric view where we can think of vectors as arrows in space. We can also think of it as map from the dual space to the real numbers.
My question is: what is the geometric viewpoint of the other tensors?
For example, a differential $k-$form is an antisymmetric type $(0,k)$ tensor. Geometrically, we can think of as measuring the flux through an infinitesimal k-parallelotope at each point of the manifold. Furthermore its exterior derivative can be thought of as measuring the net flux through the boundary of a $(k + 1)$-parallelotope at each point. https://en.wikipedia.org/wiki/Exterior_derivative
What about for an arbitrary tensor?
