Flatness over regular local rings of dimension 3

Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely generated modules.

• If $n=1$ then $M$ is flat iff $M$ is torsionfree, and the functor $M\mapsto M/(torsion)$ is left adjoint to $i$.
• If $n=2$ then $M$ is flat iff $M$ is reflexive, and the functor $M\mapsto M^{\vee\vee}$ is left adjoint to $i$.

If $n=3$ (or in fact $n\ge 3$), is there a nice characterization of flat modules? Is there a left adjoint to $i$, providing a flatification functor? Or are there reasons why such a left adjoint can't exist?