When you say extra layers of skin , I am guessing it is a bit like a scab ( which forms after an injury
over the skin heals ).
Usually this problem will occur if there is excessive drying of the environment inside the nose . If you do try to pull it out , the underlying skin will become raw and may even bleed .
This may be a mild drying which has gotten out of hand , or it may be the start of a condition known as Atrophic Rhinits
Irrespective of the cause there are a few things you should do
1. Use Saline nasal drops 3-5 times a day ( a few drops in both nostrils )
2. Avoid picking or scratching your nose
3. meet with a local ENT who can examine your nose properly
please feel free to get in touch if you have any further queries