This tensor, denoted Gμν is in a sense dual to the Maxwell Tensor in electrodynamics. It is defined as Gμν = ½εμνρσFρσ, where the tensor εμνρσ is very similar to εijk in that it is antisymmetric under any change of indicies, is zero for repeated indicies, and has ε0123=1.
Gμν 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 dμGμν=0