The idiom save your skin means to save yourself from any type of difficulties or problem without caring about others.
1. Manoj saved his skin by telling police that his friend had stolen the money.
2. Politicians save their own skin by showing that they have invested the money in favour of public instead for their own use.
3. When the plane was crashing Michael jumped from the plane with the only available parachute leaving his friend alone, he practically saved his skin.