Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation 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, but I'm not sure if going the way of geometric algebra is helpful here.