Tensor cross product proof: $Sa \times Sb = \det(S)S^{-T}(a \times b)$

Question

I need to prove

$$Sa \times Sb = \det(S)S^{-T}(a \times b),$$

given that $a$ and $b$ are vectors and $S$ is a second-order (rank $2$) tensor.

I have the hint: vectors $u=v$ iff $u\cdot a=v\cdot a$ for all vectors $a$.

I literally have no idea where to start with this so any help would be greatly appreciated :)