My understanding of the definition of a tensor is admittedly a bit shaky. As far as I know, a tensor is a multilinear map.
However,
When we say "tensor" is this what we are really referring to:
A multilinear map which maps vectors to a scalar (and not an arbitrary vector)? Arbitrary multilinear maps are then "built up" from tensors?
The cross product is referred to as a rank 1 (psuedo)tensor. Physicists like to say that it is a (psuedo)vector.
But the cross product is bilinear, and so shouldn't it therefore be a rank 2 tensor? Furthermore, the cross product is a bilinear map from two vectors to another. This doesn't match our definition above which says that a tensor maps to a scalar.
I understand that we can view "vectors" as linear maps on the dual space. Is that all that is meant by saying that the cross product is a vector?
A $(k, l)$ tensor on a vector space $V$ over the field $\mathbb{F}$ is a multilinear map $(V^*)^k\times V^l \to \mathbb{F}$.
Let $L(V_1\times \dots\times V_p, W)$ denote the vector space of multilinear maps $V_1 \times\dots\times V_p \to W$. There is an isomorphism $L(V_1\times\dots\times V_p, W) \to L(V_1\times\dots\times V_p\times W^*, \mathbb{F})$ given by $T \mapsto \widetilde{T}$ where $\widetilde{T}(v_1, \dots, v_p, \phi) = \phi(T(v_1, \dots, v_p))$.
In particular, there is an isomorphism $L((V^*)^k\times V^{\ell}, V) \cong L((V^*)^{k+1}\times V^{\ell}, \mathbb{F})$, so a multilinear map $(V^*)^k\times V^{\ell} \to V$ can be viewed as a $(k + 1, \ell)$ tensor. Similarly, (provided $V$ is finite-dimensional) there is an isomorphism $L((V^*)^k\times V^{\ell}, V^*) \cong L((V^*)^k\times V^{\ell+1}, \mathbb{F})$, so a multilinear map $(V^*)^k\times V^{\ell} \to V^*$ can be viewed as a $(k, \ell+1)$ tensor.
Example: If $\mathbb{F} = \mathbb{R}$ and $V = \mathbb{R}^3$, then the cross product defines a multilinear map $T : V\times V \to V$ given by $(v_1, v_2) \mapsto v_1\times v_2$. This can be viewed as a multilinear map $\widetilde{T} : V^*\times V\times V \to \mathbb{R}$ given by $(\varphi, v_1, v_2) \mapsto \varphi(v_1\times v_2)$. That is, we can view the cross product as a $(1, 2)$ tensor on $\mathbb{R}^3$.