Here is at least a partial answer to the question, to supplement some comments I already made. The essential case is that of a simple Lie algebra over $\mathbb{C}$. For each simple type there is a "natural" irreducible representation as well as the (irreducible) adjoint representation; these coincide just for type $E_8$. Many sources (such as Chapter 8 of Bourbaki's Groupes et algebres de Lie) specify the dimensions. For types $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$, these are respectively: $n+1, 2n+1, 2n, 2n, 27, 56, 248, 26, 7$ and $n^2-2n, 2n^2 +n, 2n^2+n, 2n^2-n, 78, 133, 248, 52, 14$.
As indicated in the question, the second (or complementary) exterior power of the natural module agrees with the adjoint module for types $B_n, D_n$. But dimension comparison seems to rule out such coincidences in other cases. In fact, higher exterior powers of the natural representation are usually not even irreducible beyond type `A_n$. (Fundamental representations overlap here somewhat, but require case-by-case discussion as done in Bourbaki.) Much is known classically about dimensions of irreducibles as well as decomposition of symmetric and exterior powers, but it can take a lot of work to make the details explicit for each simple type.