You always have a full minute to compose your answer. Quite a bit longer, in fact. You can even answer questions which were asked (and answered) years ago.
The point of the system isn't to create a race to a single answer, it's to create good and useful content. If you have content to contribute, then please by all means contribute it. (And thank you for anything you've contributed so far, sincerely.)
It's better that a question get a dozen answers than that potential answers be turned away. What if the one who locked the question isn't writing a good answer? Or an entirely incorrect answer?
The voting system, while not perfect, is in place just for this reason. The good answers bubble up to the top, bad ones sink to the bottom.
Don't be dissuaded when answering by that little notification that somebody else has already answered. Post yours anyway. You may get just much out of it as the asker does. You may be looking at the problem from a different angle, or adding a piece of information that the other answer overlooked, etc. If after answering you feel that another answer exactly matches yours and that yours shouldn't be there as a duplicate, then you can delete it. (That's actually a very selfless and community-minded thing to do, putting the quality of the overall content before your own reputation score.)
Just provide content, let the community determine what to do with it.