R^3/Cross product/Oriented orthonormal basis/Fact

Let ${}u_{1},u_{2},u_{3}$ be an orthonormal basis of ${}\mathbb {R} ^{3}$ with

${}\det {\left(u_{1},\,u_{2},\,u_{3}\right)}=1\,.$

Then the

cross product ${}x\times y$ can be computed with the coordinates of ${}x$ and ${}y$ with respect to this basis (and the formula from

Definition).

Proof, Write another proof