TITLE: How are these two ways of thinking about the cross product related?
QUESTION [25 upvotes]: 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.

REPLY [3 votes]: Geometric (Clifford) algebra provides yet another way to understand the cross product.  If a vector represents a 1D subspace containing the origin, the wedge (outer) product is a bilinear form on 2 vectors ($a \wedge b$) that represents the subspace containing both vectors.  The vector has basis $e_1, e_2, ..., e_n$ ($n$ elements), so the wedge product has basis $e_1\wedge e_2, e_1\wedge e_3, \ldots, e_{n-1}\wedge e_n$ ($\binom{n}{2}$ elements).  The cross product is a way to represent this subspace (using its dual, the normal vector).  This works only for 3D, where the basis sizes match, using $e_1 e_2 \to e_3,\, e_2 e_3 \to e_1,\, e_3 e_1 \to e_2$.