Is there a better way to write the vector identity $\nabla\times[(\vec{A}\cdot\nabla)\vec{A}]$?

Question

I have the vector expression $\nabla\times[(\vec{A}\cdot\nabla)\vec{A}]$, and I'm wondering if there's a way to write this vector identity in a better way, i.e. in terms of just single derivatives of $\vec{A}$, or in terms of divergences rather than the way it is written now.

Thanks in advance!