  We give examples of countable linear groups in $SL_{n}(R)$ for $n \ge 3$,
with no nontrivial normal abelian subgroups, that admit a faithful sharply
2-transitive action on a set. Without the linearity assumption, such groups
were recently constructed by Rips, Segev, and Tent. Our examples are of
permutational characteristic $2$, in the sense that involutions do not fix a
point in the $2$-transitive action.
