In order for Japanese to become more skillful, I am studying every day. [Syntax]

The correct answer is "上手(じょうず)になる" (jouzu ni naru). This sentence uses the purpose pattern "~ために" (in order to ~). A verb must be in its plain form to connect to "ために". "上手(じょうず)になる" means "to become skillful" and is a plain form verb phrase. Therefore, "日本語(にほんご)が上手(じょうず)になるために" correctly expresses the purpose: "in order to become good at Japanese". "日本語(にほんご)" and "が" are already in the sentence, forming the subject of the purpose clause. "ために、毎日(まいにち)" contains the conjunction and an adverb, both of which follow the verb phrase.

正解は「上手になる」です。「〜ために」は動詞の普通形に接続し、「〜するために」という目的を表します。「上手になる」は「上手になるために」と正しく接続します。「日本語」と「が」はすでに文中にあり、「ために、毎日」は「上手になる」の後ろにくるため不適当です。