Characterization of cross product (proof that only exists in dimension 3) In order to show that the cross product exists only in $\mathbb R^3$, our teacher puts this remark, "Exercise left to the reader". I have no idea how to solve it. Here's the problem:
"Let $B:\mathbb R^n \times \mathbb R^n \to \mathbb R^n$ be a surjective bilinear map and antisymmetric so that $B(u,v)=0 \iff \text{u and v are linearly dependent}$. Then $n = 3$."

Write $B_i=\sum_{jk}\varepsilon_{ijk}u_jv_k$. Hereafter, we'll drop the $\sum$. There are constraints on $\varepsilon_{ijk}$, e.g. antisymmetry ($\varepsilon_{ijk}=-\varepsilon_{jik}$) and orthogonality ($\varepsilon_{ijk}=-\varepsilon_{ikj}$). But beyond three dimensions, these leave some degrees of freedom, whereas in $3$ dimensions the symbol $\varepsilon_{ijk}$ is determined up to scaling. The former fact is why the product @runway44 mentioned isn't unique, even up to scaling.
