Are you planning to lower the entire ceiling? Just to get lower than the steel beam? If this is a hard ceiling (gypsum board) you can't cover up junction boxes, valves, or anything that needs access. To answer your question the 2x4 s are put on edge as that provides the most strength and allows longer spans. You can use metal studs with hanger wire or channel iron, or there are metal ceiling systems that look kind of like a suspended acoustical ceiling grid, except the pieces are wider and gypsum board is screwed to them.

Ok thanks Seaborn

Why are you doing this? It's a waste of time and money.

Personally I would want as high a ceiling as I could achieve in as many places as possible in a basement.

Like Mark, I would be glad to have the highest ceiling possible.

As to your proposal, you don't want the framing on the flats at all. On edge only. Go thru the pile of 2 x 4's at the lumber yard and you'll see why. Your ceiling will end up looking like the bottom of a bunch of row boats if you frame on the flat.

Thanks for the feedback. Yeah our contractor said he prefers to use metal studs instead

If you don't lower it, you might be able to do some overhead weight work, depending on your height.