If V is a vector space of dimension N, and $f\colon V \to V$ is an endomorphism, one defines its determinant to be the scalar corresponding to the induced map $\wedge^N V \to \wedge^N V$.
I like this definition, but I can't really say I know how to work with it. For example, how do you prove that det is an algebraic map? I'd like to show that GL(V) is a variety, without picking a basis for V.
Edit: As pointed out in the comments, det is a map End(V) $\to$ End($\wedge^N V$) = k, where k is the base field. This map is far from being linear, in fact by picking a basis for V one can look up in any textbook that it is a polynomial of degree N in the entries of the corresponding matrix, i.e. an element of Sym$^N$End(V)$^*$. So the question boils down to identifying det as an element of SymEnd(V)$^*$ = SymEnd(V$^*$).
It's easier just to give a coordinate-free proof that $GL(V)$ is a variety. The point is that the multiplication map
$$\text{End}(V) \times \text{End}(V) \to \text{End}(V)$$
is an algebraic map (because it's bilinear), and $GL(V)$ is the closed subvariety of $\text{End}(V) \times \text{End}(V)$ given by the inverse image of $\text{id}_V \in \text{End}(V)$ under this map.