TITLE: $x^TAx=0$ for all $x$ when $A$ is a skew symmetric matrix
QUESTION [5 upvotes]: Let $A$ be an $n\times n$ skew symmetric matrix.
Show that $x^TAx =0 \ \forall x \in \mathbb R^n$.

How to prove this?

REPLY [10 votes]: You know that $-A=A^T$, so 
$x^T A x = (x, Ax)$ (1)
but we also have 
$x^T A x = x^T (-A^T) x = -x^T A^T x = -(Ax)^Tx = -(Ax,x)$ (2)
Now notice (1) and (2) need to be the same.

REPLY [3 votes]: Hint
$$(x^TAx)^T=x^TA^Tx=-x^TAx$$
and $x^TAx$ is a real(Why?) so...