  Let $M$ be a perfect module of projective dimension 3 in a Gorenstein, local
or graded ring $R$. We denote by $\FF$ the minimal free resolution of $M$.
Using the generic ring associated to the format of $\FF$ we define higher
structure maps, according to the theory developed by Weyman in "Generic free
resolutions and root systems" (Annales de l'Institut Fourier} 68.3 (2018), pp.
1241--1296). We introduce a generalization of classical linkage for $R$-module
using the Buchsbaum--Rim complex, and study the behaviour of structure maps
under this Buchsbaum--Rim linkage. In particular, for certain formats we obtain
criteria for these $R$-modules to lie in the Buchsbaum--Rim linkage class of a
Buchsbaum--Rim complex of length 3.
