How to transform skew-symmetric matrix into vector

We can build a skew-symmetric matrix from a vector $v$ using formula
$$S(v) = \begin{pmatrix} v \times i \ \ v \times j \ \ v \times k \end{pmatrix}$$ where $i,j,k$ are vectors of the standard basis.

But how to obtain vector $v$ from its skew-symetric matrix?

Is there any linear formula for that ?

AFTER 2 hours

I think I have found solution:

$$v=-ij^TS(v)k-jk^TS(v)i-ki^TS(v)j$$

Thank you for discussion.

AFTER 20 hours

I have one additional question:

could $v$ be calculated with one formula also from $S^2(v)$ ?

AFTER 31 hours

It seems it’s hard to obtain straightforward formula.

$v$ can be obtained from $S^2(v)=vv^T-v^TvI$ but by a rather tedious analysis.

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