You could be developing an allergy to the new linen, they sometimes do not show up until you have been exposed to them for awhile.
To relive the itch, try some cortisone cream or Calamine lotion, both seem to help the irritation and calm down that horrible itching.
If all else fails, see a dermatologist and have him do some tests to determine what is taking place.
Do you own pets? If so, it may be fleas! If not, the problem may be as simple as you say about mosquito's, but it may also be bedbugs! There have been many reports about them being very prevalent this year. If it's bedbugs..a great deal of cleaning will have to be done to your bedding.
An over the counter antihistamine will help stop the itching you are suffering from, such as Benadryl or Claritin. Also you could try to use some Calomine lotion to the areas to see if that helps. There are some topical anti-itch creams available as well. Good Luck