Is there a more efficient way (than the standard) to solve the Sylvester equation $AX+XA = C$, with $A$ symmetric positive semi-definite, $C$ antisymmetric, and the solution $X$ known to be antisymmetric? (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.