Antisymmetric tensor fo rank four in three dimensional space

Question

I'm trying to find an expression for a tensor of fourth rank (in three dimensional space) that is antisymmetric with respect to its indices pairs, ie. a tensor with the following property: $$ \frac{1}{2}\left(T_{ijkl}-T_{klij}\right)=T_{ijkl} $$ Is there a particular procedure to derive such a tensor? I tried combining the tensors $\delta_{ij}$ and $\epsilon_{ijk}$ in a way to obtain the above relationship, but I am out of luck. All I can find is for constructing isotropic tensors. Speaking of which, I have another related question. Are isotropic tensors necessarily symmetric, and are anisotropic tensors necessarily anti-symmetric?