BrE - I am writing to you.
AmE - I am writing you.
BrE - Please, write to me soon!
AmE - Please, write me soon!
BrE - I'll write to you next week.
AmE - I'll write you next week.
I believe that "write to me/you" is also used in AmE. However, "write me/you" is never used in BrE.
Student or Learner