Catalysis of the reaction: 3-(alpha-D-mannosyl)-beta-D-mannosyl-R + UDP-N-acetyl-alpha-D-glucosamine = 3-(2--alpha-D-mannosyl)-beta-D-mannosyl-R + H+ + UDP.
uridine diphosphoacetylglucosamine-alpha-1,3-mannosylglycoprotein beta-1,2-N-acetylglucosaminyltransferase activity