Can one define wedge products using determinants for $n$-forms?

Question

I was talking to Ted Shifrin in math chat yesterday and he mentioned there is a way to define wedge products using determinants.

As far as I understand, given a set of vectors $x,y,z,v,u... \in \Bbb{R}^n$, the wedge product is an operation defined on those vectors such that

1. $v \wedge v = 0$
2. $u \wedge v = - v \wedge u$
3. $cu\wedge v= c(u\wedge v)$

4. $(u_1 + u_2) \wedge v = u_1 \wedge v + u_2 \wedge v $

   $u\wedge (v_1 + v_2) = u \wedge v_1 + u \wedge v_2 $

Apart from this, I really don't know much about it. I tried understanding this concepts using tensor products and free vector spaces, but since I didn't get anywhere with that, I'm still unsure what a wedge product is other than abstractly via those axioms.

This answer provides an example how to define wedge products for a 2-form: Wedge product and determinants

My Question:

1. Does the determinant definition of a wedge product work for $n$-forms?
2. If this definition using determinants is satisfactory, why do I see wedge products defined more often using free vector spaces and or tensor products?