I would screw 2x4 studs to each frame leg, then attach the ledger board on top of the 2x4's. Attach joist hangers for 2x8 floor joists. 3, 10' 2x10's as ledger board on the other side.
I might use 6x6 for the free standing posts, is I was going to store much weight up there. I've seen 4x4's that were junk.
If I had the ceiling room and room to swing down, I would make a long 1 pc. hand winch raised/lowered, staircase.
I saw one a guy made. It had 2x6 0r 8 stringers, and treads 11" deep. He was going to put a hand rail on 1 side, for safety carrying thing up and down.
When I put my sailboat in my back yard (on lawn) I built a staircase from 2x6's and it was so much easier to come and go, than with a ladder.