= Solution
For an ordinary real <quadratic form>, split the matrix into its <symmetric part> and <skew-symmetric part>:
$$
A=M+K,\qquad M=\frac{A+A^T}{2},\quad K=\frac{A-A^T}{2}.
$$
The scalar $\mathbf x^TK\mathbf x$ equals its transpose $\mathbf x^TK^T\mathbf x=-\mathbf x^TK\mathbf x$, so it is zero. Consequently
$$
\boxed{\mathbf x^TA\mathbf x=\mathbf x^TM\mathbf x,\qquad
M=\frac{A+A^T}{2}=M^T.}
$$
This uses transpose because the source form is bilinear. A complex Hermitian form $\mathbf x^\dagger A\mathbf x$ would instead require the conjugate-transpose decomposition.
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