 Solution

Dualize the antisymmetric tensor to a vector,An isotropic symmetric rank-two tensor depending quadratically on one vector can only be a linear combination of ${\delta }_{ij}{d}_k{d}_k$ and $d_i{d}_j$: after rotating $d$ to the third axis, invariance under all rotations about that axis forces the matrix to be $\mathrm{diag}(a,a,b)$.

Since $D_{ik}{D}_{jk}={\delta }_{ij}{d}^2-d_i{d}_j$ and $D_{kl}{D}_{kl}=2d^2$, the most general answer may be writtenThe two displayed tensors are independent, for example by taking $d$ along a coordinate axis and comparing the components parallel and perpendicular to it. Hence there are exactly two independent terms.