If you are talking about yourself is "escrevo". If is about you and someone else together is "escrevemos". Or, to make it easier, if the verb ends in "O" is first person singular. If ends with "mos" is 1st person plural. Eu escrevo Tu escreves Ele/Ela/Você escreve Nós escrevemos Vós escreveis Eles/vocês escrevem.
Yes, we know, and you're right, but this sentence has no context, He writes a book is grammatically ok, but it needs to be in a specific context. I agree if you had the Portuguese sentence first, the right translation should be "he's writing a book", but if you had the English sentence first, the Portuguese translation is ok I guess. For the English sentence first, it has to be reported, and the better translation "he's writing a book" to be suggested as a correction first.