I am looking for a text on linear algebra which is entirely basis free, and makes heavy use of the exterior algebra.

For instance, let $V$ be an $n$-dimensional vector space over a field $k$. Determinant of a map $\phi : V \rightarrow V$ may be defined as the map $\Lambda^n V \stackrel{\Lambda^n (\phi)}{\rightarrow} \Lambda^n V$, which is a scalar after choosing any basis of the one dimensional space $\Lambda^n V$. I am looking for a textbook which covers linear algebra in terms such as this.