The thing with differential forms is that (to me) they look a bit magical (why those sign changes? why alternate?). Only when thinking of volume forms do you begin to understand it (the determinant being the paradigm of volume element, etc.).

I think some grassmannian computations would be good in this context but, on the other hand, they would become very cumbersome very soon.

As someone says below: Spivak's Differential Calculus on Manifolds is exceptionally good.