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.
Identify the job by title and let the recipient know how you heard about it. Summarize your qualifications for the job, specifically your work experience, activities that show your leadership skills, and your educational background. Ask for an interview, stating where you can be reached and when you will be available. If your prospective employer is located in another city and you plan to visit the area, mention the dates for your trip. If you are applying for a specific job, include any information pertinent to the position that is not included in your resume.