TITLE: Why is cross product only defined in 3 and 7 dimensions?
QUESTION [28 upvotes]: Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems mysterious to me.

REPLY [6 votes]: I like Sanath Devalapurkar's explanation!
But also, after some more research, I see that if you identify $\mathbb{R^7}$ with the strictly imaginary octonions, you can explicitly define the cross product in terms of octonion multiplication with the following:
$$x \times y = \operatorname{Im}(xy)=\frac{1}{2}(xy-yx).$$ 
We can conversely construct a Euclidean space with the cross product isomorphic to the octonions. If $V$ is a seven-dimensional Euclidean space with a given cross product, we can have bilinear multiplication on $\mathbb{R} \oplus V$ like this:
$$(a,x)(b,y)=(ab-x \cdot y, ay+bx+x\times y)$$
forming an isomorphism $\psi:\mathbb{R} \oplus V \to \mathbb{O}$.
We can do the same thing in three dimensions, and in any $n-1 > 2$ dimensions such that an division algebra over $\mathbb{R}$ exists for $n$ dimensions - so cross product is defined only $3$ and $7$ dimensions.