[math] \newcommand{\mathds}{\mathbb}[/math]

Prove with full details, based on the above, that the determinant of the systems of vectors

[[math]] \det:\mathbb R^N\times\ldots\times\mathbb R^N\to\mathbb R [[/math]]

is multilinear, alternate and unital, and unique with these properties. Then try to prove as well this directly, without any reference to geometry.