Is there a way (more efficient than the standard vectorization) to solve the following [Sylvester equation][1] in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive semidefinite, and the matrix $C$ is skew-symmetric? Does this fact about $X$ follow from the statement?

**Background:** the matrices $C$ and $X$ are really [bivectors][2], but I'm not sure if going the way of geometric algebra is helpful here.


  [1]: https://en.wikipedia.org/wiki/Sylvester_equation
  [2]: https://en.wikipedia.org/wiki/Bivector