Yup. Any short story (in our case sudden fiction, which is a story with fewer than 1,000 words) should be complete - with a beginning, a middle and an end. Unless that part of your novel is a complete story in itself (and why would it be?), no short story market is going to publish it.
You can check the
Index for Sudden Fiction Markets thread (or for longer short stories, the
Publishing Short Fiction - Markets over in the short stories forum) for markets to send your story to.
There are paying and non-paying markets, it's up to you where you submit it. But resources like
Duotrope or
Ralan will be able to tell you which is which.
Some magazines like cover letters, some don't. Some require a short bio, some don't. Some have deadlines and submission periods, some don't. You need to check the submission guidelines for any of the magazines you're interested in to find out which does which.
Don't forget -- you can always post the story
here for critique if you want to make sure it's polished and ready to go!
Good luck and happy submitting.