Since $x^T A x \in \mathbb{R}^{1 \times 1}$, then $x^T A x$ is symmetric due to
R3987: 1-by-1 matrices are always symmetric. That is, $(x^T A x)^T = x^T A x$. Applying
R3747: Transpose of finite product of real matrices, we then have
\begin{equation}
x^T A^T x
= (x^T A x)^T
= x^T A x
\end{equation}
Subtracting $x^T A^T x $ from each side, we get $x^T A x - x^T A^T x = 0$. Thus
\begin{equation}
x^T \left( \frac{A - A^T}{2} \right) x
= \frac{1}{2} \left( x^T A x - x^T A^T x \right)
= \frac{1}{2} \cdot 0
= 0
\end{equation}
$\square$
