In formal British English, do not say "write
me", or "she wrote him". These are fully acceptable in American English; but in formal British English we prefer to say "write to me", and "she wrote to him." The only things we can write are texts, words: one can "write a letter", or "an assignment".
This is a good example (in the British dialects of English) of the difference between a Direct and an Indirect Object. The Direct Object is the thing that one writes, 'puts on paper': the letter or assignment. The Indirect Object is the person for, or to, whom one writes.
- "I wrote her [Indirect Object] a letter [Direct Object]"
- "I wrote my tutor [Indirect Object] an assignment [Direct Object]."