fuv={{0,b3,-b2,-I e1},{-b3,0,b1,-I e2},{b2,-b1,0,-I e3},{I e1,I e2,I e3,0}}
MatrixForm[fuv]

guv=ConstantArray[0,{4,4}];
Do[Do[sum=0;
Do[Do[sum=sum+LeviCivitaTensor[4][[m,n,i,j]]*fuv[[i,j]],{i,1,4}],{j,1,4}];
guv[[m,n]]=I sum/2,{m,1,4}],{n,1,4}]
MatrixForm[guv]

LeviCivitaTensor[4][[1,2,3,4]]
LeviCivitaTensor[4][[1,2,4,3]]
LeviCivitaTensor[4][[4,1,2,3]]
LeviCivitaTensor[4][[1,2,3,1]]