= Axial vector of an antisymmetric second-rank tensor
{title2=$a_i=\tfrac12\epsilon_{ijk}A_{jk},\quad A_{ij}=\epsilon_{kij}a_k$}
In three-dimensional Euclidean space, contracting an <antisymmetric second-rank tensor> with the <Levi-Civita symbol> gives an <axial vector>. The epsilon contraction identity proves the inverse formula: $\frac12\epsilon_{kij}\epsilon_{klm}A_{lm}=\frac12(A_{ij}-A_{ji})=A_{ij}$. With this index convention,
$$
A=\begin{pmatrix}0&a_3&-a_2\\-a_3&0&a_1\\a_2&-a_1&0\end{pmatrix},\qquad Ax=x\times a.
$$
Thus $A$ is minus the <cross-product matrix> for $a\times x$. Proper rotations transform $a$ as a vector; an orientation-reversing orthogonal change adds a determinant sign. Stating the index convention prevents a sign ambiguity when converting an antisymmetric tensor into a cross-product operator.
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