For $SO(4)$, there are two independent quadratic Casimir operators. In the defining basis where the Lie algebra elements are antisymmetric matrices $\omega_{ab}$ with two $4-\text{indices}$, these are given by
$$ C_2 = \omega_{ab} \omega_{cd} \delta^{ac} \delta^{bd}, $$ $$ C'_2 = \omega_{ab} \omega_{cd} \varepsilon^{abcd}. $$
I'm looking for similar formulas for $SO(5)$. The first one is clear:
$$ C_2 = \omega_{ab} \omega_{cd} \delta^{ac} \delta^{bd}, $$
but the second expression doesn't generalize in a straightforward way to $\varepsilon$ with $5$ indices.
What's the explicit expression for $C'_2$, the second independent quadratic Casimir of $SO(5)$?