Let $A\in\mathbb{R}^{n\times n}$ be a skew-symmetric matrix, is the Rayleigh quotient of $A$ always $0$?

Question

Let $A\in\mathbb{R}^{n\times n}$ be a skew-symmetric matrix, is the Rayleigh quotient of $A$ always $0$?

The Rayleigh quotient is define as following: $$\mathcal{W}_{\mathbb{R}}(A)=\big\{ \frac{x^*Ax}{x^*x}:x\in\mathbb{R}^n\setminus\{0\} \big \}$$