'What about Eskimos' is a good question, and you're not the first to ask it. One important question to ask is why the lack of sunlight leads to lighter skin. It's not something that happens magically in response to sunlight. Rather, lighter-skinned people must have some advantage at higher latitudes which means that genes causing lighter skin are more likely to be passed on.
One of the best hypotheses as to what this advantage is relates to vitamin D. Most of our vitamin D is synthesised internally, but this process requires sunlight. We need to synthesise vitamin D because unlike, for example, vitamin C, it's not common in our diets.
Most of our diets, anyway. Eskimos have traditionally had rather different diets to most people, a diet rich in fish and fat, blubbery meats. Or, to put it another way, a diet rich in vitamin D.
In support of the idea that sunlight is the driving selective force behind skin tone, it's been noted that the correlation between skin colour and UV exposure is extremely strong. The main exceptions to this correlation are all populations that have migrated in historical times.
You ask as well about people of 'Mongolian race', but people in North East Asia do have light skin. What they don't typically have is blonde hair and blue eyes but, as Razd pointed out, these are caused by different mutations than light skin, and they are unlikely to be connected to sunlight. A mutation causing blonde hair arise entirely independently in tropical Melanesia, where you can find people with dark skin and blonde hair.