This tensor, denoted G_μν is in a sense dual to the Maxwell Tensor in electrodynamics. It is defined as G_μν = ½ε_μνρσF^ρσ, where the tensor ε_μνρσ is very similar to ε_ijk in that it is antisymmetric under any change of indicies, is zero for repeated indicies, and has ε⁰¹²³=1.
G_μν is then also antisymmetric, and has components:

(0  -B1 -B2 -B3)
(B1  0  -E3  E2)
(B2  E3  0  -E1)
(B3 -E2  E1  0 )

That is, it is just like the Maxwell tensor, with B in place of E and -E in place of B. Maxwell's Equations imply that d_μG^μν=0