3-4 months is too short of a time to ASK for a promotion but definitely not too short of a time to start looking at how to get one. Eyes on the prize!
Do you have an official job description you can review? If so, read that and see if you're doing everything on it. If you're not, work to add those things into your normal job functions. If there is a job description for the next job up from you (office manager maybe?), read that one too and see which of those things you could start doing (without stopping on others' toes) or start learning about.
If you have skill gaps, look into classes you could take to build them. Does your company have any education reimbursement?Come up with a plan for taking classes during work time and after hours so you can propose both ways with your boss and see what flies.
I would suggest initiating a sit-down meeting with your boss in order to ask for feedback about how you're doing. A good way to ask is "am I meeting your expectations?" If Yes... "what would I need to do to exceed them?" You could initiate the meeting by sending them an Outlook invite and in the body saying those are the things you'd like to discuss. For now, don't mention the promotion desires (unless they ask where you'd like to be blah blah), just talk about wanting to excel at your current position and make their life easier. It sounds like you're already doing that.
I really admire your focus on being good at what you're doing, your desire for feedback, etc. You sound like someone I would be happy to hire. :)