The Lie algebra of a group acts on itself through the commutator $ T_a \in ad$:
$$ T_a \circ T_b = [T_a,T_b] \in ad $$
I assume the same should be true if we have an antisymmetric adjoint, as for example for $\mathfrak{so}(n)$, because [Antisymmetric, Antisymmetric]=Antisymmetric.
The adjoint can be written as tensor product representation of the fundamental and the antifundamental $f \otimes \bar f$ or $f \otimes f$ if $f$ is real. For $\mathfrak{so}(n)$ the adjoint is simply the antisymmetric part of this tensor product. I tried to compute the corresponding transformation laws here, using this, but failed miserably.
What is the correct transformation law for the symmetric part of the tensor product repreentation $f \otimes f$? A naive guess would be
$$ T_a \circ S = [T_a,S], $$
because [Symmetric,Antisymmetric]=Symmetric.