There exists a notion of generalized form: cochains. These are linear functionals on a certain class of currents; the problem is that you would like to put on your class of currents a meaningfull topology (maybe induced by a norm) and you would also like this class to be stable under boundary.
Smooth $k-$surfaces with boundary have the second property, but not the first one. In order to put some topological structure on them, you have to enlarge the space and consider integer rectifiable currents, where you have a norm-induced topology, compactness theorems and stability under the boundary operator.
The cochains have been defined on flat currents (which contain the previous ones); you could look up for flat cochains into Federer's Geometric Measure Theory (4.1.21 or somewhere further). 4.1.19). They ultimately are functionals $\ell$ associated to a couple of measurable forms $(\alpha,\beta)$ of degree $k$ and $k-1$ so that for every flat current $T$ you have
$$\ell(T)=T(\alpha)+ \partial T(\beta)=T(\alpha)+T(d\beta)$$
They were also studied by Whitney, I believe you can find something in his Geometric Integration Theory (Chapter 9 and following).
I don't know if this was what you were looking for, but that's what I know about dualizing currents.