I was reviewing Levi-Civita symbols and came across this identity:
$$ \epsilon_{ijk} \epsilon_{ijn} = 2 \delta_{kn}$$
My first thought was the identity that involves a determinant:
$$\epsilon_{ijk}\epsilon_{lmn}=\det\left| \begin{array}{cccc} \delta_{il} & \delta_{im} & \delta_{in} \\ \delta_{jl} & \delta_{jm} & \delta_{jn} \\ \delta_{kl} & \delta_{km} & \delta_{kn} \end{array} \right| $$
which is frequently used to prove other identities, such as $$\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$
If I were to employ this approach, I would take the determinant and replace $l$ with $i$ and $m$ with $j$, which is obviously easy to do. However, I seem to recall there being a significantly more elegant approach that doesn't resort to using this determinant-based definition or the identity that follows - does anyone remember what it is?