  Let $R$ be a commutative ring. For any projective $R$-module $P_0$ of
constant rank $2$ with a trivialization of its determinant, we define a
generalized Vaserstein symbol on the orbit space of the set of epimorphisms
$P_0 \oplus R \rightarrow R$ under the action of the group of elementary
automorphisms of $P_0 \oplus R$, which maps into the elementary symplectic Witt
group. We give criteria for the surjectivity and injectivity of the generalized
Vaserstein symbol and deduce that it is an isomorphism if $R$ is a regular
Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$
over a perfect field $k$ with $c.d.(k) \leq 1$ and $6 \in k^{\times}$.
