$T_{ij} = S_{ij} + \varepsilon_{ijk}V_k$ where $S_{ij}$ is symmetric. I need to find $S_{ij}$ and $V_k$ in terms of $T_{ij}$.
I found $S_{ij}$ by swapping i,j in the original equation and then adding the two together, but I am unsure on how to find $V_k$. What exactly is the 'inverse' of $\varepsilon_{ijk}$?
EDIT: Think I've made some progress. Applying $\varepsilon_{ijq}$ to both sides gave me $\varepsilon_{ijq}T_{ij} = \varepsilon_{ijq}S_{ij} + 2\delta_{kq}V_k \implies \varepsilon_{ijq}T_{ij} = \varepsilon_{ijq}S_{ij} + 2V_q$. Then swapping i,j and adding the two equations gave me $4V_k = \varepsilon_{ijk}(T_{ij} - T_{ji})$. Can this be simplified further? (Also is it correct?)