Best motivation for an antisymmetric product to get the area between vectors

Question

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?

Answer 1

Here's one strategy for motivation: start with $\Bbb R^2$, keeping in mind that $\wedge^2 \Bbb R^2 \cong \Bbb R$ and the antisymmetric product can be thought of as signed area (a notion that students will be familiar with from calculus. Noting that this computation is a determinant, we can explain the antisymmetric nature of area computation with the argument given here (see everything below "How does that follow from our abstract definition?"). It might be helpful to bring attention to the fact that giving the area a sign allows us to make this function (multi-)linear.

For $\Bbb R^3$, abstract this to the cross product. We could justify the fact that this scalar now becomes a vector quantity by saying that the natural generalization of assigning area a "sign" is assigning the area a "direction" corresponding to the orientation of the plane. Again, adding this structure allows us to compute area "linearly".

From there, for the purposes of motivation, it should suffice to say that the pattern continues. To justify the number of entries in the vector, we could say that we need enough parameters so that a unit vector can be used to specify the orientation of the plane. If you like, you could go into a bit more detail about the Plücker embedding; if you look online, you can find many different explanations including some that are geared towards computer science students.