Skew-symmetric matrices dot product condition

Skew-symmetric matrices dot product condition

From the Wikipedia page on skew-symmetric matrices:
Denote with $\langle\cdot,\cdot\rangle$ the standard inner product on
$\mathbb{R}^n$. The real $n$-by-$n$ matrix $A$ is skew-symmetric if and
only if $\langle Ax, y\rangle = -\langle x, Ay\rangle$ for all $x,y\in
\mathbb{R}^n$.
I can't see how this follows from the definition $A^T=-A$ for
skew-symmetric matrices.