Can we define random forcing through Baire space?

Question

We can talk about Cohen real both in $2^\omega$ and $\omega^\omega$ since all countable forcing notions are equivalent to $\mathbb{C}$, while in random forcing I've only seen random real in $2^\omega$. So, my question is：
1.In the random forcing definition, can we replace $2^\omega$ by $\omega^\omega$?
2.If 1 is wrong, can we talk about random real in $\omega^\omega$ by other ways?