Looks good, but it doesn't translate to "did I do something wrong?". It sounds more like someone accused you of doing something wrong, and you're responding: "What did I (in particular, as opposed to everyone else) do wrong?" The emphasis is on you. I think 나 잘못했나? (Did I make a mistake?) is better

I do not recommend you write "내가 뭐 잘못했는데?" in written expression. It could/might give the impression to the reader that you are putting his/her pecker up because of the potential intonation. It implies that you think you did nothing wrong. Therefore the safe way is to change some words: "내가 뭐 잘못했나 봐(요). 난 잘 모르겠는데(요)." Informal(formal). Or, "내가 뭐 잘못한 것 있으면, 말해 줘(요)." "내가 뭐 잘못한 것 있어(요)?. 있으면 말해 줘(요)." + "난 잘 모르겠어(요)." Good luck! ^^

"내가 뭘 잘못했는데?" means "What did I do wrong". "내가 뭔가 잘못했니?" is "Did I do something worng?" you were looking for. 뭔가/무엇인가 = something. 니/나/나요 is the proper conjugation dedicated to "whether" questions.