Reddit reviews Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics)

I haven't read it myself, but I've heard good thing about V. I. Arnol'd's Mathematical Methods of Classical Mechanics. I don't know how far he goes in generalizing from R^n to arbitrary configuration spaces, but since he does use the language of differential geometry in his treatment, the book might be what you're looking for.

For an elementary account, there is a book by Weintraub called Differential Forms, which is very accessible. It is explicitly written as a "complement to vector calculus".

For a more advanced treatment take a look at Arnold's Mathematical Methods of Classical Mechanics. There is a standalone chapter on differential forms which is extremely well-written and well-motivated (as is the rest of the book, for that matter).

I will remember that. I have Arnold's Mathematical Methods of Classical Mechanics