Let $R$ be a commutative ring. Let $G=(g_{ij})$ be a skew-symmetric matrix with $g_{ij}\in R$. Then if $C$ is an $n\times n$-matrix in $R$, then $\rm{Pf}$$(C^{\top}GC)$ = $\rm{det}$$(C)$$\rm{Pf}$$(G)$.
Several proofs of the statement over $\mathbb{Z}$ establish $\rm{Pf}$$(U^{\top}TU)$ = $\pm\rm{det}$$(U)$$\rm{Pf}$$(T)$ using $\rm{det}$$(G)=\rm{Pf}$$(G)^2$, take off the negative sign by replacing $U$ with identity, and finally replace $U$ with $C$ and $T$ with $G$. But in doing so, there's the assumption that $n\times n$ coefficients $u_{ij}$ be algebraically independent --- why is this assumption crucial here?