TITLE: Cross product in higher dimensions
QUESTION [36 upvotes]: Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear.
Suppose instead we have two vectors $x$ and $y$ in $3$-space. Then the cross product gives us a new vector $x \times y$ that's orthogonal to the first two. Furthermore, cross products are bilinear.

Question. Can we do this in higher dimensions? For example, is there a way of turning three vectors in $4$-space into a fourth vector, orthogonal to the others, in a trilinear way?

REPLY [2 votes]: Yes, and apart from other answers an interesting approach to think about it is using Clifford's algebra.
This can introduce you the basic concept in a nonrigorous but approachable manner.
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/