Using the definition of tensors as tensor products, let $\operatorname{CROSS}$ be a type $(1,2)$ tensor defined by $$\operatorname{CROSS} =\sum_i\sum_j\sum_k (e_k \cdot (e_i \times e_j))\,\,e_i^*\otimes e_j^*\otimes e_k$$
which is a linear combination of other tensors, therefore a tensor.
Observe that $\operatorname{CROSS}_{ij}^k u^iv^j = (u \times v)^k$. Hence, the cross product actually is a tensor. ([EDIT] Wikipedia agrees).
What do people mean when they say the cross product is not a tensor? And what do they mean by "the" cross product when they say that? Because the above proves it is.
[edit] I understand on an intuitive level what this article on pseudo-vectors is trying to say about the cross product. But I don't understand how to formalise that "problem". I'm not talking about formalising the "solution", which involves exterior algebra or Clifford algebra.
([edit] Attempt at formalising: The examples they give don't involve changing the coordinate system, but involving transforming the actual physical system by some reflection. For instance, if in the cars example, the coordinate system were reflected in the way they described, then the angular momentum vector would point in the opposite direction, contrary to what they're saying. More generally, if by reflecting a left-handed cross product, you end up with another left-handed cross product, then your reflection was not a change of basis, but a change that affected physical objects.)
See here for people claiming that the cross product is not a tensor. It seems like what's really happening is that they're defining the cross product to be the exterior product. In other words, when they say cross product, they don't actually mean cross product.
Also this article on Levi-Civita symbols seems to be saying something relevant.
