$x^TAx=0$ for all $x$ when $A$ is a skew symmetric matrix
Solution 1:
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.
Solution 2:
Hint $$(x^TAx)^T=x^TA^Tx=-x^TAx$$ and $x^TAx$ is a real(Why?) so...