So, partially due to my own ignorance, I was under the impression that there is no cross product in $n$-dimensional space except for where $n = 3$ and $n = 7$. I mean, I've read that a lot, from many different sources.However, the cross product in $n$-dimensional space is just the Hodge star operator where $k = n - 1$. That is, the $3$-D cross product takes in two (e.g. $n - 1$) $3$-D vectors and outputs one $3$-D vector. Where $n = 7$, the cross product takes in six $7-D$ vectors, and