Also, possible "your arm is red" because "χέρι" means both hand and/or arm.
Yes, if you need you can say...αντιβράχιο = forearm, βραχίονας/μπράτσο = upper arm,
καρπός = wrist, αγκώνας = elbow.
But for both hand and arm it's "χέρι" unless you need to be more specific they you use one of the above.
There is also παλάμη but that's only "palm" not the whole "hand".
Does this have some idiomatic meaning? Perhaps like the English expression "caught red-handed"? Or does it only mean that the hand is literally red?