I've never heard of $K_2(R)$ having a description as a more easily/elementarily described object attached to $R$. Borel showed that $K_2(R)$ is torsion. The order of $K_2(R)$ appears in "Theorem" 31 of Soule's Soulé's notes on higher algebraic K-theory of rings of algebraic integers where $K_2(R)$ (I think that "Theorem" 31 is called the Quillen-Lichtenbaum conjecture).
I've never heard of $K_2(R)$ having a description as a more easily/elementarily described object attached to $R$. Borel showed that $K_2(R)$ is torsion. The order of $K_2(R)$ appears in "Theorem" 31 of Soule's notes on higher algebraic K-theory of rings of algebraic integers where $K_2(R)$ (I think that "Theorem" 31 is called the Quillen-Lichtenbaum conjecture).