Which is the best way to motivate to **undergraduate students** that the area of the parallelogram of two vectors $v,w\in\mathbb{R}^n$ is given by taking the modulus of some "antisymmetric product"?

For instance, if $n=4$, how would you (geometrically?) motivate / prove that the area between $v=(a,b,c,d)$ and $w=(e,f,g,h)$ is the modulus of some

$$(af-be,ag-ce,ah-de,bg-cf,bh-df,ch-dg)\in\,\,???$$

What would you say about the nature of this "antisymmetric product"? (it is indeed $\bigwedge^2\mathbb{R}^n$, but that's not for undergraduate students)

Perhaps one may argue that one needs to know the area of the projections onto axis planes (e.g. $af-be$) to recreate the area in the bigger space?

Would this reasoning be extensible to motivating the $k$-volume of $k$ vectors in $\mathbb{R}^n$ as the modulus of, again, some antisymmetric product whose components are $k$-determinants of the original coordinates?