In my time, I had to do a lot of vector calculus so I learned a few tricks. I would rewrite the equation in components, and express vector product using the Levi-Civita symbol. There is just one basic identity to remember:
\[ \epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{jl}\delta_{im}\]
With this and:
\[\partial_i(v_mv_m) = 2v_m \partial_i v_m \]
you can derive the identity you were using.
