。
I wrote down his name so I wouldn't forget.
Literal
Forget-not [so-that-ように] his name [object-を] noted-down.
Subject is omitted — the speaker is implied by default. The ~ないように + 書き留めた frame ('wrote down so as not to forget') describes the purpose of a memo. Whether to spell out the subject in Japanese is largely about context: if the surrounding discourse already establishes who's doing the writing, the pronoun can drop without ambiguity.