Cross product and right hand rule

Question

Is there a simple proof that the cross product (defined as the usual determinant) always obeys the right hand rule?

Answer 1

By the linearity and anticommutativity, it suffices to prove that $i\times j=k$. $$\left|\begin{array}{ccc}i&j&k\\1&0&0\\0&1&0\end{array}\right|=k$$

A similar computation proves that $j\times k=i$ and $k\times i=j$.