For the sake of : for the purpose or benefit of someone or something; to satisfy the demands of someone or something. For one's sake : for one's good or benefit; in honor of someone However when you use "For others", it may mean either for the sake of others or for the use of others. e.g. "Give up a little for others"may mean "give up a little for the sake of others" or "give up a little so that the others can use it". e.g. "Donate a little for my sake" = "To donate a little in order to appease Me / satisfy me / make me happy / in my honour". Here, I may/may not use your donation. It might be the same as "Donate a little to the beggar for my sake" "Donate a little for Me" = It may also mean that I am going to use whatever you are going to donate. However "donate a little to the beggar for me" may mean either to donate something to the beggar to make me happy or to donate something to the beggar on my behalf. So whether or not it is good to use them interchangeably depends on the context.

Another usage is "for the sake of it," which means to do something that is not really necessary, just for the sake of doing it. For example, an editor may be editing an article and find that there is really nothing wrong with it, but they might make a few changes just for the sake of making changes, or just for the sake of it. This means that they feel as though they should make changes, because that's their job, even if in this example there was no need to make any changes.