Alternative to Axler's "Linear Algebra Done Right"

If Linear Algebra Done Right doesn't work, then try Linear Algebra Done Wrong, by Sergei Treil. This seems to meet both of your requirements.

If you want a higher level textbook, allow me to suggest "Advanced Linear Algebra" by Steven Roman.

The text is primarily concerned with abstract vector spaces, but it does treat matrices in detail and uses them when it is natural to do so. It will probably also cover all the linear algebra you need in an undergraduate degree. You can read samples of it here:

http://books.google.com/books?id=FV_s8W58D4UC&hl

http://www.amazon.com/Advanced-Linear-Algebra-Graduate-Mathematics/dp/0387728287

I'm a fan of the book by Hoffman and Kunze. It's a standard, concise, proof-based text.