In the cases that you decide skipping class is worth it, here's a good guide to politely skipping class.
you are not subtle with excuses
If there isn't a legitimate reason to skip class, don't make up a reason. Lecturing professors have heard it all before, easily sorting honest reasons from lame excuses. While you may think a late night of DODA can be covered for with "a family emergency," your prof will probably be able to see through you.
Don't make an excuse. If you feel the need to approach the professor, just earnestly apologize for missing class and ask if there's anything you can do to make up the work. If you plan to miss class, ask a friend to take good notes for you and email the prof asking for the work you'll miss.
legitimate reasons have paper trails
For legitimate reasons to miss class, get a paper trail. If you're very sick, go to the doctor. Although this might be a pain, a doctor's note can be turned in to your professor to excuse an absence. As silly as it might be, an email from your parents describing the family difficulty can turn a recalcitrant professor into a sympathetic mentor. If you go to a GRT or other dorm staff for help due to drama that's really affecting you, ask that person to email your professor. While they may not be a grand authority, their word holds more weight than yours on its own.
Check to see if your school has a department or office for students experiencing difficulties. For example, students with crutches or arm braces can go to the Office of Disabilities to get transportation to and from class and class note takers, respectively. Or a student dealing with exam anxiety can go to Student Support Service to request more time on an exam or a separate room for test taking.
safe skip: the lecture after an exam
This is a favorite lecture of mine to skip, especially since I'm usually still tired from studying for the exam. A portion of the lecture is "wasted" going over the exam results . I say waste since most classes post solutions to exams online and the next office hours are dedicated to answering questions on the exam. The start of new material is very important as it is the base for everything else, but it's also the easiest to learn. So I find this to be an overall decent lecture to miss.
leave at break
If you show up for class but feel the need to leave midway through, the best time to leave is at the break. If there's no break, try to aim for a natural pause in discussion or speach. This is something that should be saved for the worst most desperate case as most professors will notice and may even take you leaving personally.
do not leave if exodus
Some lectures suffer from a large mass of people leaving all at once. In the case that a group of people are leaving, don't join them. Staying on the day that the class is nearly empty is equivalent to telling the prof that you value their class. Professors notice your presence then more than ever.