Is there an existing linear mapping that maps a 3-dimensional vector:
$$\mathbf{v}=\begin{pmatrix} v_1\\v_2\\v_3 \end{pmatrix}$$
to a corresponding skew-symmetric matrix:
$$\mathbf{V}=\begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0\end{pmatrix}$$
A tensor of order 3 should probably be defined.
Edit The question is related to the following one:
knowing that there exists a matrix $\mathbf{V}\in\mathbb{R}^{3,3}$ such that for a given vector $\mathbf{v}\in\mathbb{R}^3$:
$$\forall\mathbf{x}\in\mathbb{R}^3,\quad\mathbf{V}\mathbf{x}=\mathbf{v}\times \mathbf{x}\quad\Leftrightarrow\quad \mathbf{V}=\mathrm{CPM}(\mathbf{v})$$
where CPM means cross-product matrix, can we express in a frame-invariant fashion the quantity:
$$ \mathbf{V}_\mathrm{A}=\mathrm{CPM}(\mathbf{Av})$$
where $\mathbf{A}$ is any $3\times 3$ real matrix?
(the result is $\mathbf{V}_\mathrm{A}=(\mathbf{VA})^T-\mathbf{VA}+\mathrm{tr}(\mathbf{A})\mathbf{V}$ but was obtained by calculating each coordinate of the left-hand and right-hand side matrices and subsequently identifying each term)
Best Answer
The name of the tensor you're looking for is the Levi-Civita or permutation tensor. In cartesian coordinates, $\epsilon_{ijk}$ is equal to $+1$ for any even permutation of 123 and $-1$ for any odd permutation.
The permutation tensor represents a directed unit volume. As such volumes have different expressions in other coordinate systems, the Levi-Civita tensor will depend on the coordinate system used. However, it represents a fundamental, coordinate-system invariant object called the pseudoscalar of the space. And there are ways to use the pesudoscalar without resorting to the tensor approach, avoiding the problem of how the pseudoscalar has different expressions in different coordinates.
The pseudoscalar enforces the notion of duality. In 3d space, vectors are dual (orthogonal to, or normal to) planes. In some circles, we call directed planes bivectors. Using the pseudoscalar in this way converts back and forth between vectors and their dual bivectors. The skew-symmetric rank-2 tensor (matrix) you have here is the direct representation of such a bivector.