Prove $x^TAx = 0$ $\implies$ A is skew-symmetric We know for a skew-symmetric matrix A, $x^TAx = 0$. But is the converse statement true, i.e. does $x^TAx = 0$ imply A is skew-symmetric? If yes, then how to prove it?

$x^T A x = 0\ \forall x$ can be written as: 
$\sum_i \sum_j x_j a_{ij} x_i = 0\ \forall x$
That is:
$x_1^2 a_{1,1} + x_1 a_{1,2} x_2 + ... = 0$
The only way to have $x_i x_j$ cancel out $x_j x_i$ is by: $a_{ij} = -a_{ji}$, and the remaining terms $x_i^2 a_{ii}$ must be zero (for all $x$), so we must have $a_{ii}=0$. Therefore $A$ is antisymmetric.
