Quick Tip: Don't use your work email address for job searching. Use your personal email account, or set up a unique account to use just for your job hunt. Many free online email services exist, like Gmail and Yahoo mail, that you can use to set up a new email account exclusively for your job search.
How do you know for sure whether or not to use a closing? One good test is to ask yourself whether this email is more akin to an instant message/text or a business letter. If you're giving your teammate a quick update on an ongoing project, a formal closing might not be necessary; if you're throwing your hat in the ring for a promotion, it definitely is required. When all else fails, and you're still not sure, err on the side of caution and include it. You'll never go wrong by being too polite and respectful.