I am given to understand that $\bigwedge^2(\mathbb{C}^4)$ is not irreducible, as a complex representation of the group $U(4)$ of $4 \times 4$ unitary matrices. My explanation is that $\dim_\mathbb{C} \left( \bigwedge^2(\mathbb{C}^4) \right) = 6$, whereas I seem to have read in section 23 of these notes that the irreducible representations of $U(4)$ are of the form $(\det)^n \otimes \operatorname{Sym}^k(\mathbb{C}^4)$. In particular, the possible dimensions of irreducible representations would be $\binom{4+k-1}{k}$ which takes values $1,4,10,\ldots$.
Question: Supposing I am not labouring under a misapprehension here, do we have a conceptually clear splitting of $\bigwedge^2(\mathbb{C}^4)$ into irreducible representations?