I would suggest you look into Ranked and Clan Battles. Ranked uses Random Matchmaking, but it functions which an exclusionary system. Each league can only see others that are within the same league (qualifiers see the "next league" as you are auditioning for the part). While this isn't hard-and-fast MMR, it does have a refinement effect which yields higher skilled games on average with each league climbed.
Clan Battles functions in an MMR-like way as you win rating or lose rating through your battles. Your team will eventually find the spot it's going to float in which is indicative of your team's skill relative to the rest.