$\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{a}$ (commutative)

$\mathbf{a} \times \mathbf{b}$ is parallel to both $\mathbf{a}$ and $\mathbf{b}$

$\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$

Cross product is defined in all dimensions