It isn't bad for you per se, but it is healthier to eat breakfast every day. Breakfast act as a jumpstart for your body, so it gets you digesting what you ate faster and burning calories faster. This, in turn, makes you feel more awake when you do eat breakfast, and can also help you lose weight.
Yes... It is probably the worst thing you can do to your body and your brain. Your body needs fuel to give you energy and to make everything work properly. If you skip breakfast you can guarantee that you will have lots less energy and you will not be as mentally productive as you normally might be. That is a politically correct way to say that NOT eating breakfast is pretty stupid. It can lead to all sorts of health problems.