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.

Probably the narrow question here can be studied for classical types (the Lie algebras or associated simply connected compact Lie groups) in a concrete way, but ultimately the "correct" approach requires comparison of highest weights of the various irreducible representations involved. For this one should check the "planches" at the end of Bourbaki's Chapter 6 for the way the highest root is expressed in terms of fundamental weights, etc. I'm not sure whether any single source gives a concise account of both the concrete and abstract representation theory: standard, adjoint, and fundamental representations, along with a description of the exterior powers of the standard module.

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. (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.

Probably the narrow question here can be studied for classical types (the Lie algebras or associated simply connected compact Lie groups) in a concrete way, but ultimately the "correct" approach requires comparison of highest weights of the various irreducible representations involved. For this one should check the "planches" at the end of Bourbaki's Chapter 6 for the way the highest root is expressed in terms of fundamental weights, etc. I'm not sure whether any single source gives a concise account of both the concrete and abstract representation theory: standard, adjoint, and fundamental representations, along with a description of the exterior powers of the standard module.

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.

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.

Probably the narrow question here can be studied for classical types (the Lie algebras or associated simply connected compact Lie groups) in a concrete way, but ultimately the "correct" approach requires comparison of highest weights of the various irreducible representations involved. For this one should check the "planches" at the end of Bourbaki's Chapter 6 for the way the highest root is expressed in terms of fundamental weights, etc. I'm not sure whether any single source gives a concise account of both the concrete and abstract representation theory: standard, adjoint, and fundamental representations, along with a description of the exterior powers of the standard module.