I am having difficulty showing the following relation with respect to orthonormal basis $\{{\bf{e}}_{1},{\bf{e}}_{2},{\bf{e}}_{3}\}$:
$$ \epsilon_{ijk}{\bf{S}}_{jk} = \epsilon_{ijk}{\bf{W}}_{jk}\tag{a}\label{a} $$
where $\bf{S}$ is a tensor and $\bf{W}$ is the skew tensor component of $\bf{S}$. It is also given that the axial vector of $\bf{S}$ is ${\bf{w}} = \text{axl}\bf{S}$ and is defined by:
$$ {\bf{W}\bf{u} = \bf{w} \times \bf{u}} \tag{b}\label{b}$$
I believe I understand it at a qualitative level. I know the following:
- The cross product of two vectors is an axial vector. The scalar components of the cross product are the scalar components of an axial vector.
- The scalar components of the cross product can be expressed as the product of a skew matrix (i.e. the scalar components of a skew tensor) and a vector.
- The scalar components of the cross product can be expressed in the following ways using indicial notation \begin{align} ({\bf{a}} \times {\bf{b}})_i &= \epsilon_{jki}a_jb_k \quad \text{(3x1 matrix of the scalar components of the cross product)}\tag{1}\label{eq1}\\ ({\bf{a}} \times {\bf{b}})_{ij} &= a_ib_j - a_jb_i\tag{2}\label{eq2} \end{align}
where Eq.~\eqref{eq2} is a 3x1 matrix of cross product scalar components represented as the product of a skew matrix (3x3) and a vector (3x1).
Given all the above, Expression \eqref{a} makes sense when I think of it in terms of the scalar components of each side of the equation. The left side is the product of a skew matrix and a vector and the right side is simply the scalar components of the cross product of $\bf{w}$ and $\bf{u}$.
Where I struggle is finding a starting point for Expression \eqref{a}. Since I don't know where to start I will begin by stating what the left side, $\epsilon_{ijk}{\bf{S}}_{jk}$, represents. This quantity is a compact way of stating the scalar components of a vector, represented by the following 3x1 matrix: \begin{bmatrix} S_{23} - S_{32}\\ S_{31} - S_{13}\\ S_{12} - S_{21} \end{bmatrix}
This looks awfully familiar to expressions I have seen before of the scalar components of the cross product. In fact, if $S_{jk} = w_j u_k$ I would say this is equal to the scalar components of the cross product of $\bf{w} \times \bf{u}$. However, I do not know why $S_{jk}$ would be equal to $w_j u_k$ and I am failing to see how this helps me show my original objective.
I would appreciate any hints or comments on what I have stated thus far.