The k-currents are defined as dual space to the spaces of all smooth k-forms. (These monsters are used to work with the minimal k-surfaces.)

Assume I want to look at the generalized k-forms; they can be defined as certain functionals on the space of all smooth k-surfaces with boundary.

*Did anybody consider such "generalized k-forms"?*

In fact I am looking for such generalized 2-forms on $\mathbb R^n$ with values $so(n)$, this should be some kind of generalized curvature. (Instead of integral I should consider parallel translation around the boundary. If $n=2$, it gives me a sign-measure.) But I am also interested in the linear case.