ad-invariant skew-symmetric form $\omega(x,y,z)$

Question

Let $\mathfrak{g}$ be a lie algebra and $(,)$ a symmetric bilinear ad-invariant form on $\mathfrak{g}$, and define $\omega(x,y,z) = ([x,y],z)$, for $x,y,z \in \mathfrak{g}$. Show that $\omega$ is skew-symmetric and ad-invariant. For ad-invariance, I note that $\omega(ad(x),ad(y),ad(z)) = ([ad(x),ad(y)],ad(z)) = (ad[x,y],ad(z)) = ([x,y],z)= \omega(x,y,z)$, showing ad-invariance because $(,)$ is ad-invariant. How do I show skew-symmetry? I am not sure what I even have to show. I know a skew-symmetric bilinear form is one that satisfies $(x,y) = -(y,x)$, but what is meant in this context?