Inverse of the sum of a symmetric and a skew symmetric matrix

Question

Consider the following matrices $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times n}$, with $A=-A^\top$, $B=B^\top\leq0$.

From several analytical calculations it turns out that

$$ (A+B)^{-1}=C+D, $$

with $C=-C^\top, D=D^\top\leq0$.

1. Can anyone provide a proof for this to hold for any n?
2. Is there any analytical formulation for the matrices C and D?

EDIT: If this is of some relevance or help in proof, I might be interested also in the case that B is strictly negative.

Thanks for your help.