You can do this but consider these things:
1. Joists may be 19+ o.c. instead of 16 or 24. If so, that makes hanging drywall a little more work since the sheets won't fall on a stud. You'll have to cut and tape more.
2. You won't have the top plate to nail the drywall onto. Not a big deal, but something to consider.
3. I'd still put in fire blocking. If you should have a fire, that cavity becomes a chimney - allowing the fire to spread farther and faster.
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