Let $A\in R^n$, A is a skew-symmetric matrix if and only if $x^TAx=0, \forall x\in R^n$

Proof:

Necessity: if A is a skew-symmetric matrix, then $A^T=-A$. Hence $(x^TAx)^T=x^TA^Tx=-x^TAx$. Note $(x^TAx)^T=x^TAx$, hence $x^TAx=-x^TAx=0$.

Sufficiency: if $x^TAx=0, \forall x\in R^n$, then $x^T(A+A^T)x=0$. I think we need the lemma that if A is a symmetric matrix, then $x^TAx=0, \forall x\in R^n$ if and only if $A=0$. Since $A+A^T$ is a symmetric matrix, from $x^T(A+A^T)x=0$ we know $A+A^T=0$. Hence A is skew-symmetric.

QED

EDIT: The proof of sufficiency above is not correct. See here for a correct proof.

The below is a simpler proof.

Since $x^TAx=0, \forall x$,

$e_i^TAe_i=0$, so $A_{ii}=0$;

$(e_i+e_j)^TA(e_i+e_j)=0$, so $A_{ij}+A_{ji}=0$.

Hence $A=-A^T$.