Based upon the Zohar many women have the custom to cover their hair while sleeping (Zohar Parshas page 239, see Yalkut Shemonei Bereishis), although the covering will usually fall off.
Exposing Partial Hair:
The Pri Megadim, Mishna Berura, and Chazon Ish all conclude that a skirt must therefore cover at least from the knee and above. Tosfos writes that the shok is actually the lower part of the leg in people. The description of the term shok by the Bach (O.C. 75) as a place that if not for the chiddush din or ervah we would not consider prohibited to be exposed because it is normally covered in dirt seems to point to the lower foot as the area being discussed, not an area above the knee. Based on these sources many poskim (HaRav Vozner; the Chazon Ish also debates the proofs for this position) argue that a skirt should be much longer and cover far below the knee as well.
Of course it must covered in all physical positions she will be in during the course of the day.