The adjoint representation of $\mathrm{SO}(n)$ is the exterior product of two fundamentals.
In some cases this is an irreducible representation, e.g. for $n=3$ it is actually equivalent to the fundamental thanks to the invariant tensor $\epsilon_{ijk}$.
In some other cases it is reducible, e.g. for $n=4$ it is the direct sum of two irreducible representations $A_+ \oplus A_-$. Where a tensor in $A_\pm$ satisfies the (anti)self-duality condition $$ M_{ij} = \pm\, \epsilon_{ijkl} M_{kl}\,. $$ For what other values of $n>4$ is the adjoint representation irreducible?