I want to define a function $f$ mapping an integer $n$ to $\underbrace{\mathbb{R}^3\times\mathbb{R}^3}_n$.
Is it correct to express the function $f$ as $$ f : \mathbb{N} \to \mathbb{R}^{3\times\mathbb{N}} ~ ? $$ For me, it is some weird. I want to define a function $f:n\mapsto\mathbb{R}^n$ in a form of domain to domain. Is there a way to express the function $f$ well?
For example, $f(3)$ gives three $3$-vectors, $f(10)$ gives ten $3$-vectors, and so forth, where the $3$-vector implies a vector having $3$ elements.
For a detail example, $f(2)=\{(1,2,3), (10,5,3)\}$.
Well, that depends on how you want to treat things.
In general, you can talk about $X^{<\Bbb N}$ as the collection of finite sequences from $X$, which then you can choose to formalize as a subset of $X^\Bbb N$ where every function is eventually $0$ (you can even declare the first coordinate of each function as "its length", to avoid confusion (although it might introduce other difficulties).
So it seems to me that $f\colon\Bbb{N\to(R^3)^{<N}}$.