Solving antisymmetric tensorial equation

Perhaps

f = RandomInteger[{-1, 1}, {4, 4}]; 
Solve[
 And @@ Join[
   Thread[Equal[Flatten[Table[
       Sum[f[[l, i]] s[j, l, k] - f[[l, j]] s[i, l, k], {l, 4}], 
   {i, 4}, {j, 4}, {k, 4}], 2], 0]], 
   Flatten@Table[s[i, j, k] == s[j, i, k], {i, 4}, {j, 4}, {k, 4}]]]