There are a few ways of doing this, 1 way is what you have in your situation... 2nd-way entry to the basement is done from the exterior if there is already a set of stairs from the house that lead into the basement and the 3d-way as what you looking to do.

That said, it's hard to say without seeing what you enquiring to do because a lot of things must be taken into consideration, i.e what space you have to work with, how the sub-frame layout is, make sure there will be no headroom issues, landing clearances, etc. and in most cases doing something like this, you will need architectural drawings and a building permit so talking to a local Architect or even a contractor to make onsite evaluation is your best bet, this is not the right place for that...

But on the general note, anything can be reconfigured and reconstructed for as long as you have the budget to do it.

Good luck