In my matrix analysis course, we are seeking to understand the idea behind determinants. In class, my professor mentioned that
"The determinant of an $n \times n$ matrix can be thought of as an alternating $n$-linear function of its column vectors."
For clarity, an $n$-linear form is alternating if $x_i=x_j \Rightarrow f(x_1, \ldots, x_n) =0$ for $i\neq j$.
This idea is one that I can't quite wrap my head around. I understand that the determinant can be thought of as a scaling factor for the volume generated by basis vectors, but beyond that, I'm struggling to see how determinants relate to multilinear maps. I am also stuck on why the alternating condition is important.
I found this question on MSE, but it only confused me more.
This question provided a little more insight, but I feel like I still don't have all the prerequisite knowledge to effectively understand everything.
Any help would be appreciated.