I have to solve the problem $b=(A-A^T)x$ for $A\in\mathbb R^{n\times n}$, where $b$ and $x$ are known.

When we need to solve $Ax=b$, with $A$ not invertible, a known result is $x=A^\dagger b$ where $A^\dagger$ is the Moore-Penrose pseudoinverse. In this case I do not care about minimizing the norm of $A$ of any such thing - I just want some $A$ that satisfies $b=(A-A^T)x$. Apart from solving a feasibility optimization problem, is there a known analytical result for this problem?

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    $\begingroup$ This is still a linear system in the elements of $A$. You can find that linear system and solve it using the psuedo-inverse. $\endgroup$ – NicNic8 Feb 19 '18 at 2:51

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