As many people before me I became interested in alternating and differential forms and naturally I tried to look for some geometrical interpretation for the objects. I failed to find anything that would really explain their geometric nature, so instead I came up with three questions, which if being answered would greatly help with that.
First of all, consider a linear space $V$ of dimension $n$ with a dot product $(\cdot,\cdot)$ and the space $\Lambda^k(V)$ of $k$-linear alternative forms on $V$. The question is whether the application of $\omega \in \Lambda^k(V)$ and $x_1,\dots,x_k \in V$ can be written invariantly in terms of geometrical operations only (e.g. scaling, projection, orientated volume). It is easy to see that in case of $k \in \{1,n-1,n\}$ the answer is positive. It is positive as well, if the form is decomposable, that is can be represented as a wedge product of 1-forms. Note, that standard coordinate expression for $k$-form is not the representation in question, because it is a sum and is not invariant.
Secondly, in case that first question has no positive answer, it is natural to try to see $k$-forms as some sort of orientated $k$-volume on $V$. Consider a set $\{\omega_L \in \Lambda^k(L)| L \subset V -\;$ k-dimensional subspace$\}$. Define a map $\omega_*$ by $\omega_*(x_1,\dots,x_k) = \omega_{<x_1,\dots,x_k>}(x_1,\dots,x_k)$. Putting restrictions on $\{\omega_L\}$ the map $\omega_*$ can be made smooth. The map $\omega_*$ that we defined perfectly suits the notion of "some sort of orientated $k$-volume on V". Moreover, it can be extended on manifolds the same way as differential forms are extensions of alternate $k$-linear forms. However, it is understood that $\omega_*$ is not necessarily a $k$-form on $V$. So the question is what conditions should the set $\{\omega_L\}$ satisfy for $\omega_*$ to be a $k$-form on $V$ and whether they have any geometric meaning.
Finally, if none of previous question has an answer, then I would like to ask for examples of differential forms of dimension $k \notin \{1,n-1,n\}$ naturally occuring in mathematics or physics. Maybe they will allow me to gain more understanding in the topic.
Sorry for the english and/or if something I have written is unclear.
Thanks.