Skew-symmetric matrix/form

Question

What's the relation between a matrix $A$ that represents a skew-symmetric form, and a skew-symmetric matrix $B$?

The matrix $A$ is such that for all vectors $v,w$, we have $v^TAw=-w^TAv$.

The matrix $B$ is such that $B^T=-B$.

So their characterizations are different. Is there a relation between these two sets of matrices?

Answer 1

There is no difference. Observe:

$$\vec{v}^T{A}\vec{w}=\vec{w}^T{A}^T\vec{v}$$

This is true for all vectors $\vec{v}$, $\vec{w}\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times{n}}$.