A somewhat advanced but still physics oriented exposition can be found in Section 5.4 of
- Geometry, Topology and Physics, Mikio Nakahara, CRC Press.
Another approach, still very physics oriented, but a bit longer and slightly less andvanced can be found at Chapter 18 of
- Modern Geometry - Methods and Applications: Part I, B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Springer.
The book itself is very good, it builds everything from the start and I strongly recommend it for a first approach to geometry for physicists.
If you need something quick and to the point you might also like Section 2.8.2 of
- D-Branes, Clifford V. Johnson, CUP.
Even though the book talks about $D$-branes, the section I pointed you to is self contained and understandable from someone that does not have that kind of background.
There is also Section 8.8 of
- Quantum theory of Fields, Volume 1, Steven Weinberg, CUP.
which is really short but talks about $p$-form electrodynamics, so it might be interesting to look at.