Note 1 to entry: The components of an antisymmetric tensor are such that $T_{ij}=-T_{ji}$, and in particular $T_{ii}=0$.

Note 2 to entry: An antisymmetric tensor defined on a three-dimensional space has three strict components which can be considered as the coordinates $W_1\text{,}{W}_2\text{,}{W}_3$ of an axial vector:

$\left(\begin{array}{ccc}0& W_3& -W_2\\ -W_3& 0& W_1\\ W_2& -W_1& 0\end{array}\right)$

The axial vector associated with the antisymmetric tensor $U\otimes V-V\otimes U$ is the vector product of the two vectors.