Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way?

A quick search reveals that they are, but yet the outer product of two column vectors in $ \mathbb{R}^3$ is a 3x3 matrix, not another column vector. What's the link? Thanks!

Solution 1:

Cross product is much more related to exterior product which is in fact a far going generalization.

Outer product is a matricial description of tensor product of two vectors.

Solution 2:

In Geometric algebra, the cross-product of two vectors is the dual (i.e. a vector in the orthogonal subspace) of the outer product of those vectors in $\mathbb{G}^3$ (so in a way you could say that the outer product generalizes the dot product, although the cross product is not an outer product). This is very well explained in the great introduction series to Geometric Algebra by Alan Macdonald. The relation between the cross product and the outer product is mentioned in Geometric Algebra 4 at min 6:20.

Now Geometric algebra is just Clifford algebra, with a different emphasis, so that these definitions are equivalent to the definitions in that algebra.