Of course you can say “I” in a resume! It is YOUR resume!
It is perfectly acceptable to write your resume in the first person. You can start sentences with “I,” especially the professional summary, although you need to be careful not to start every sentence with “I”.
It is unnecessary to start every responsibility with “I” and make your bullet points sound awkward and repetitive. Hiring managers know that you are talking about yourself when referring to an accomplishment or responsibility on your resume. You don’t need to start sentences with first-person pronouns like I, me, us, my, or we.
For example, rather than writing “I managed all office correspondence,” you can state, “Managed all correspondence.” Recruiters and hiring managers claim that doing this makes a resume much easier and less tedious to read.