 Skew-symmetric part of a matrix

Every real square matrix decomposes as $A=S+K$, where $S=(A+A^T)/2$ is its symmetric part and $K=(A-A^T)/2$ is a skew-symmetric matrix. For a real column vector $x$, $x^{T}Kx$ equals its transpose $x^T{K}^{T}x=-x^{T}Kx$, hence vanishes. Thus a real quadratic form $x^{T}Ax$ depends only on $S$.