It's pretty common to hear about the inner product between two functions, something like
$$\langle f, g\rangle = \int{f(x)g(x)dx}$$
This can measure how "aligned" the two functions are, or "project" one function onto another, just like the regular dot product between finite-dimensional vectors. But what about the exterior/wedge/cross product? As in
$$(f\wedge g) (x,y) = f(x)g(y) - f(y)g(x)$$
I saw a post about the tensor product, but I don't think it's the same thing. The exterior product between finite-dimensional vectors gives an oriented area / bivector / 2nd-rank anti-symmetric tensor. Which incidentally can also be thought of as a generator of a rotation. But it's kind of hard to imagine how to translate that concept to functions of a continuous variable? And if there is a meaning or application of it, could you then take the exterior product with a 3rd function and so on to get higher dimensional "volumes" just as with vectors?