Happy New Year, guys!

This is my first post here, hope I'm doing this right...Sooo, I am half Japanese, wasn't born there, but grew up and lived there for the first years of my childhood, and since my parents didn't plan on fully raising me there, they didn't bother teaching me Japanese. All I know is my mother language and English, and I don't think I have any Japanese documents with my name on it on me.

I have a Western and Japanese name, and although I know how to write my surname, I'm still not sure about my first one. It's "Yuji" (or "Yūji", I'm not sure lmao), but since it's a really common first name, I'm not sure about the writing. My mom kept telling me rhat it meant second child, since I'm symbolically her 2nd son, but I can't find any combinations with this specific meaning and it's kinda embarassing for me not knowing how to write it. Can someone please help me finding out how to write my first name?

tl;dr: need help finding out how to write the name "Yuji"/"Yūji" in kanji (with the meaning 2nd child)