Comment by omnicognate

The 4th edition of Linear Algebra Done Right has a much improved approach to determinants themselves (still relegated to the end, where it should be). From the list of improvements:

> New Chapter 9 on multilinear algebra, including bilinear forms, quadratic forms, multilinear forms, and tensor products. Determinants now are defned using a basis-free approach via alternating multilinear forms.

The basis-free definition is really rather lovely.