I'm new to FT, so I apologize if the answer to this question is somewhere else. I searched, but couldn't find it.

I have purchased 4 r/t business class tickets ORD-ATH on Sky Team for a family trip in Sept. ORD-FCO-ATH on Alitalia on the outbound, and ATH-CDG-ORD on AF on the return. I don't have status on any airline at the moment. We fly often, but I haven't managed to line up enough long haul trips with the same alliance to qualify for anything that matters.
I have seen a few posts about family FF accounts being available on a few airlines. Is there anywhere I could credit these flights that would allow me to earn credit towards status for the flights I buy for my children? Even if not possible with Skyteam, is it possible elsewhere? In which case I would book my future trips somewhere else. At present, I have to be the least efficient frequent flyer imaginable. Buying 4 tickets at a time I usually pick what is cheapest/most convenient, and end up with 4 people with 15-20k EQM on 3 different airlines, which gets me nothing. Thanks for any help.