My family originated in and around Slonim and the original name is Slonimsky, though sometimes it is incorrectly written as Slominsky (on my father's birth certificate, for one).  I suspect most families with the name Slonimsky came from that area, as "sky" means "from."  That said, I know a Solomon family that changed their name to Sloan, so anything is possible, I suppose. 
--
Barbara Sloan
Conway, SC