Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite matrices. Let me start by recalling a definition from [1].


> **Definition (*$D$-skew-symmetric matrix*).** A matrix $A\in\mathbb{R}^{n\times n}$ is said to be *$D$-skew-symmetric* is there exists a diagonal matrix $D\succ 0$ such that $(A-\mathrm{diag}(A))D$ is skew-symmetric.


Now, let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues $\{\lambda_i\}_{i=1}^n$ such that $\mathrm{Re}(\lambda_i)<0$. 

> ***My Question:*** Is $A$ orthogonally similar to a $D$-skew symmetric matrix with non-positive diagonal?

My question is motivated by the fact that in the $2\times 2$ case it is possible to prove that the answer is in the affirmative. 

**Proof for $n=2$.** After an orthogonal diagonalization of the symmetric part of $A$, we can assume that $A$ is in the form
$$
A=\begin{bmatrix}a_{11} & a_{12} \\ -a_{12} & a_{22} \end{bmatrix},
$$
where $a_{ij}\in\mathbb{R}$, $a_{11}<0$, $a_{11}+a_{22}<0$. Further, since $A$ has eigenvalues with negative real part, $\det(A)=a_{11}a_{22}+a_{12}^{2}>0$. We distinguish two cases:

 1. If $a_{22}\le 0$, then $A+A^{\top}\preceq 0$. Hence, $A$ is $D$-skew-symmetric with negative diagonal for $D=I$.
 2. If $a_{22}> 0$, then $A+A^{\top}\not \preceq 0$. Define
$$
	T:=\frac{1}{\sqrt{a_{22}-a_{11}}}\begin{bmatrix} \sqrt{-a_{11}} & -\sqrt{a_{22}}\\ \sqrt{a_{22}}& \sqrt{-a_{11}} \end{bmatrix}
$$
and note that $T$ is an orthogonal matrix. It holds 
$$
	T^{\top} A T = \begin{bmatrix} a_{11}+a_{22} & a_{12}+\sqrt{-a_{11} a_{22}}\\ -a_{12}+\sqrt{-a_{11} a_{22}}& 0 \end{bmatrix}.
$$
Define
$$
	D:=\begin{bmatrix} 1 & 0\\ 0 & \frac{a_{12}-\sqrt{-a_{11} a_{22}}}{a_{12}+\sqrt{-a_{11} a_{22}}} \end{bmatrix},
$$
where $D\succ 0$ since $a_{11}a_{22}+a_{12}^{2}>0$. Thus, $T^\top AT$ is $D$-skew-symmetric with non-positive diagonal.

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[1] Kaszkurewicz, E., and L. Hsu. "On two classes of matrices with positive diagonal solutions to the Lyapunov equation." Linear algebra and its applications 59 (1984): 19-27.