The following is stated in a text I am using regarding rotations of cartesian tensors: A Cartesian tensor of rank 2 is a dyadic formed out of two vectors $U$ and $V$. One simply takes a Cartesian component of $U$ and a Cartesian component of $V$ and puts them together $$T_{ij} := U_{i}V_{j}.$$ Cartesian tensor like this is reducible-that is, it can be decomposed into objects that transform differently under rotations. Specifically, for the dyadic above we have $$U_iV_j = \frac{U \cdot V}{3} \delta_{ij} + \frac{(U_i V_j - U_j V_i)}{2} + \bigg( \frac{U_i V_j + U_j V_i}{2} - \frac{U \cdot V}{3} \delta_{ij } \bigg)$$
Question: How does it follow that the second term can be written as $\epsilon_{ijk}(U \times V)_{k}$ and how does this imply that it has 3 independent components?
Thanks for any assistance.