I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner. I now know, not one, but *two* ways of doing this, and I can't quite see how they're related:

- The cross product is the Lie bracket in the Lie algebra of $\text{SO}(3)$.
- The cross product is the Hodge star map $\Lambda^2(V) \to V$ where $V$ is an oriented $3$-dimensional real inner product space.

Okay, so there's one obvious relation here: $V$ has automorphism group $\text{SO}(3)$. But for some reason I can't figure out where to go from here. A good starting point would be to exhibit a canonical isomorphism between an oriented $3$-dimensional inner product space $V$ and the Lie algebra of $\text{Aut}(V)$. Maybe this is obvious. In any case, I would appreciate some clarification.