'Write me' is only acceptable in US English. In British English 'write to me/wrote to me' is the only correct way to write it.

Both are acceptable, but it is better to insert the "to". Without the "to", it sounds a little bit hillbillyish.

Thank you for the comment!

Write me a letter. Write a letter to me. (If there's a word between "write" and "me", "to" is used).