Does あなたに手紙をかきますmeans I am writing a letter for you?
I,ve seen that diologue
お母さんに電話をしますか。
はい、します
I think the basic pattern is A は B に something を movement
right ?
so in the diologue あなたは is omitted,because it is clear.
Can I omit 私は in my statement also?
thanks!

Question

Does あなたに手紙をかきますmeans I am writing a letter for you?
I,ve seen that diologue
お母さんに電話をしますか。
はい、します
I think the basic pattern is A は B に something を movement 
right ?
so in the diologue  あなた は is omitted,because it is clear.
Can I omit 私は in my statement also?
thanks!

Answer

Yes, you are right.
You can also omit the subject 私は 彼は　それはetc.
So the sentence （私は）あなたに手紙を書きます means I will write a letter for you.