Suppose we are working in $\mathbb{R}^3$ and have the defined the usual scalar product and proven Cauchy-Schwarz. Then we can define the angle, $\theta$, between non-zero vectors by requiring $a\cdot b= |a||b| \cos(\theta)$. (We assume we have defined $\cos$ analytically and proven all the usual properties.)
If I then define the cross product, $a\times b$, using the usual coordinate formula how do I prove that $|a\times b| = |a| |b| \sin(\theta)$?