The Mishna (Shabbat 12:4) discusses the prohibition of writing on Shabbat. In that context, it says the following:
הכותב על בשרו - חיב. המסרט על בשרו - רבי אליעזר מחיב חטאת ורבי יהושע
One who writes upon his skin is obligated [to bring a sin offering].
Concerning one who engraves his skin, Rabbi Eliezer obligates him to
bring a sin offering but Rabbi Yehoshua exempts him.
In the gemara (Shabbat 104b), the mishna there says that it's the sages (and not Rabbi Yehoshua in particular) who exempt him from bringing a sin offering, but no reason is given. In the Rambam's peirush, he suggests that it's because Rabbi Yehoshua (or "the sages" in the Bavli's version) did not consider engraving on the skin to be a form of writing. If it's not a legitimate form of writing then by doing it on Shabbat one is only transgressing a rabbinic prohibition.
It seems to me that, following this logic, writing on one's skin in ink (which is what the former part of the mishna speaks about) is a legitimate form of writing. One might prohibit it on the basis of mar'it ha'ayin, but I've not seen such a prohibition ever spelt out.