I need to show that for an antisymmetric cartesian tensor $\Omega_{ij}$, $$ \Omega_{il} x_l x_j - \Omega_{jk} x_k x_i = 2 K_{ik} \Omega_{kj}$$ where $K_{ij} = x_i x_j$ and $x_i$ are position coordinates.
I realise $\Omega$ is antisymmetric and $K$ is symmetric, but I cant use these properties to get to the result, unless I assume $T_{ij} = \Omega_{il} x_l x_j$ is antisymmetric in $ij$. I know this is probably basic but I'm stuck. Any hints would be much appreciated :).