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

(Copied from my own comments.) 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 for $0\leq k\leq n$ we have $$\bigwedge^k V_1 = \bigoplus_{0\leq\ell\leq k,\;\ell\equiv k\pmod{2}} V_\ell$$ (no multiplicities). A possible reference is: : Fulton & Harris, Representation Theory: A First Course (Springer GTM 129), theorem 17.5.