Prove that $\int_0^1 {\bf v \, . \, W}(s)\: {\bf v} \, ds = 0$ for skew matrix $\bf W$ and vector $\bf v$

Question

I'm at a loss as to how to show that

$\int_0^1 {\bf v \, . \, W}(s)\: {\bf v} \, ds = 0$

For any vector ${\bf v}$, where ${\bf W}(s)$ is skew (symmetric) at every point in $s$.

This assertion is from page 310 of Continuum Mechanics by C.S. Jog

Answer 1

For any skew-symmetric matrix $W$ and vector $v$, we always have $v\cdot Wv = 0$, since $v\cdot Wv = W^\top v\cdot v = -Wv\cdot v = -v\cdot Wv$. (So the integration is a red herring.)