Would it make sense/work to do something like this ?

  (a x b )(c x d) =>     ???

  o       o             o   o
   \     /               \ /    
    o - o         =>      o
   /     \               / \
  o       o             o   o

Would the circle in the middle of the cross be a tensor of order 4 then?

Edit: It's been a while since I did that but wouldn't that be the following in sum notation?

  eps_ijk a_j b_k eps_ilm c_l d_m = (new tensor)_jklm a_j b_k c_l d_m