Answer: The Mishna (Makkos 21a) teaches that there are two components of the prohibition of tattoos, scratching the skin and marking it with ink. If one just marked it with ink one has not transgressed the aveira (See Shulchan Aruch YD 180:1). Rambam (Avoda Zara 12:11) writes that if one only wrote but didn’t scratch the skin, one is pattur. Usually, when Rambam writes this it means that it is nonetheless prohibited miderabanan.
The Minchas Chinuch (253:1) quotes Tosafos (Gittin 20b) and the Beis Shmuel (EH 124:16) who write regarding a slave with a divorce document tattooed onto him, that one mustn’t sign their signature onto their skin as doing so is assur mideraban. R’ Shmuel Wosner (Shevet Halevi 3:111:1) challenges this, however, writing that the Beis Shmuel was referring to when one had scratched one’s skin. If one were to fill an existing scratch with ink, one would transgress an issur miderabanan though writing alone wouldn’t be assur.
R’ Avraham Avraham (Nishmas Avraham YD 180:1) notes that as the Shulchan Aruch, Chochmas Adam (89:11), Kitzur Shulchan Aruch (169:1) and Ben Ish Chai (Masei 2:15) make no mention of writing on one’s skin being prohibited, it mustn’t be an issue. R’ Shlomo Zalman Auerbach and R’ Chaim Kanievsky, likewise, maintain that one may write on one’s arm, etc.
In conclusion, there is no halachic issue with writing on one’s hand.