It seems like SO is dedicated to questions, but one thing that would be useful would be for people to share something they learned. E.g. Recently I had an issue where I discovered IIS 7 is locked down so that even URL encoded strings which in turn contain an Ampersand are blocked from access. To fix it required a Registry entry change, something I discovered only was referenced in two obscure places on the internet. While I could have asked the SO community and waited for a bounty, one other option other than answering my own question would be to post the problem and solution. I think there are community questions and wikis, but it may be useful to have a third element called something like Lessons Learned or something.
I think the usual (and if I remember correctly, officially encouraged) way is in fact asking the question, and answering it by oneself.
As earlz points out, this is not a very widespread practice. That, however, is no reason not to put valuable content onto SO. I would say if your "monologue" question gets closed because people don't like your answering your own question, take the case here on Meta. If it's a good, valid question, you will most likely get good feedback here, the question reopened, and a meta link preventing future closures.
It is perfectly fine to document the issue you solved by asking a question and answering your own question. Just keep the time between the question and answer short (e.g. prepare it in advance). You can also leave a comment to your question explaining why you answered your own question and perhaps add a reference to one of the articles below.
See the answers to these questions: