Catalysis of the reaction: GDP-alpha-D-mannose(2-) + a (mannosyl)2-(N-acetylglucosaminyl)2-diphosphodolichol = H+ + GDP(3-) + a (mannosyl)3-(N-acetylglucosaminyl)2-diphosphodolichol.
GDP-Man:Man2GlcNAc2-PP-dolichol alpha-1,6-mannosyltransferase activity