Calculation of cross product square in formula

I have a difficulty with the formula for the power radiated by a charge with arbitrary motion,

\begin{equation} P_{rad} = \frac{q^2 c \gamma^6 \mu_0}{6\pi}\Big[\dot{\boldsymbol{\beta}}\cdot\dot{\boldsymbol{\beta}}-\big(\boldsymbol{\beta}\times\dot{\boldsymbol{\beta}}\big)^2\Big]. \end{equation}

I do not understand how you can have the square of a cross product $\big(\boldsymbol{\beta}\times\dot{\boldsymbol{\beta}}\big)^2$, as the cross product is a vector, not a scalar. How do I explicitly compute the square of this cross product?