Comment by Gro-Tsen on Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations

Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial representation, then $\bigwedge^k V_1 = \bigoplus_{0\leq\ell\leq k,\;\ell\equiv k\pmod{2}} V_\ell$ (no multiplicities) for $0\leq k\leq n$. This is certainly well-known, but sadly I don't have a reference.