Yes, BrE uses 'write TO' where 'write' itself has no explicitly specified object (although 'letter' is naturally implied), whereas AmE omits the preposition in this case. When, however, the verb has an explicit object, BrE follows AmE usage in treating the verb as ditransitive.
He wrote me a letter.
would be the standard form for the sentence in either variety of English, although BrE would also accept
He wrote a letter to me.
where there was some special emphasis on the recipient.