What is the definition of $\mathbb R^n$? What is the definition of $\mathbb R^n$ space ? The set of the $n\times 1$ matrix, where coefficients are in $\mathbb R$ ?
Or, the set of mappings from $\{1,2,...,n\}$ to $\mathbb R$ ?
 A: $\newcommand{\Reals}{\mathbf{R}}$Caveat: This answer is based on my experiences with other mathematicians and is intended to fairly represent the views of the (large) majority, but is written as first-person opinion to avoid putting words in others' mouths (particularly because the second section below may be controversial to those concerned with foundations).

To answer the question at face value: If pressed for a specific Cartesian product, I'd define $\Reals^{n}$ inductively by
$$
\Reals^{1} = \Reals,\qquad
\Reals^{n+1} = \Reals^{n} \times \Reals^{1}\quad n \geq 1,
$$
"tacking on the new coordinate at the end". Consequently, a typical element of $\Reals^{n}$ would be $((\cdots((x_{1}, x_{2}), x_{3}), \dots), x_{n})$. 

As the comments and existing answers suggest, I take a less ontological, more phenomenological viewpoint in practice, focusing on how elements of $\Reals^{n}$ behave rather than on what elements of $\Reals^{n}$ are as sets.
To make an analogy with object-oriented programming, the "public interface" of the Cartesian vector space $(\Reals^{n}, \oplus, \odot)$ consists of precisely the following:


*

*An ordered collection of $n$ real-valued functions $(\pi_{j})_{j=1}^{n}$ (the Cartesian coordinates or coordinate projections) such that if $x$ and $y$ are arbitrary elements of $\Reals^{n}$, then $x = y$ (as elements of $\Reals^{n}$) if and only if  $\pi_{j}(x) = \pi_{j}(y)$ for all $j = 1, \dots, n$.

*A mapping $\oplus:\Reals^{n} \times \Reals^{n} \to \Reals^{n}$ satisfying
$$
\pi_{j}(x \oplus y) = \pi_{j}(x) + \pi_{j}(y)\quad\text{for all $x$, $y$ in $\Reals^{n}$.}
$$

*A mapping $\odot:\Reals \times \Reals^{n} \to \Reals^{n}$ satisfying
$$
\pi_{j}(c \odot x) = c\pi_{j}(x)\quad\text{for all $x$ in $\Reals^{n}$.}
$$
Using $\oplus$ and $\odot$ to denote vector addition and scalar multiplication distinguishes these operations from the field operations of the real numbers; in practice, they are universally denoted by $+$ and either $\cdot$ or simple juxtaposition.
The philosophical point about phenomenology comes down to:
Any question that cannot be phrased in terms of the public interface is not (to me) a question about the Cartesian vector space.
Similarly, any question that cannot be phrased in terms of the first item alone is not (to me, in the sense I work with $\Reals^{n}$ in daily practice) a question about $\Reals^{n}$, but a meta-question.
By contrast, you're asking about ontology, namely, about the "private data" or "implementation".
To implement $\Reals^{n}$ in set theory, the inductive construction in the preceding section suffices. Happily, the order of grouping doesn't matter, because (as noted in the comments) the Cartesian product of sets is associative up to a canonical bijection. In other words, we can, if we wish, change the implementation details (regroup the Cartesian product) without touching the public interface (without altering "how elements of $\Reals^{n}$ behave" as mathematical objects or as vectors).
It's alternatively possible to implement $\Reals^{n}$ as the set of real-valued functions on the set $\{1, 2, \dots, n\}$, with $\pi_{j}$ denoting evaluation at $j$. Some mathematicians do this. One pedagogical advantage is that function spaces are immediately seen to be natural generalizations of $\Reals^{n}$.
Coda: These are good technical issues to iron out (+1). That said, in practice, I expect most non-foundational mathematicians view this type of distinction in the same light as the following quip of Woody Allen, on the fundamental material constituent of the universe, which I offer as a good-natured jest:

Democritus called it  atoms. Leibniz called it monads. Fortunately the two men never met, or there would have been a very dull argument.

A: Let $n \in \mathbb{N}$. Then the simplest definition of $\mathbb{R}^{n}$ may be 
$$
\mathbb{R}^{n} := \{ (x_{1},\dots, x_{n}) \mid x_{1},\dots,x_{n} \in \mathbb{R} \},$$
i.e. the set of all the $n$-tuples of real numbers. If you know of Cartesian product, then you see that $\mathbb{R}^{n} = \mathbb{R} \times \cdots \times \mathbb{R}$, the $n$-fold of $\mathbb{R}$.
Note that it is customary to call a set a space if it is endowed with some structures such as a topology or a metric.
A: The following is how I conceive the idea of a Cartesian product.
Given a set $X$, we define an $m-tuple$ of elements of $X$ by the map, $$ x:\{1,...,m\}\to X$$ where the image $x(i)$ in $X$ is denoted by $x_i$. We usually represent the map by $(x_1,...,x_m)$.
So, the set of all $m-tuples $ of  $X$ is the set of all maps $x$ as defined above.
Let $\{A_1,...,A_m\}$ be a family of sets and let $X=A_1\cup ...\cup A_m$. 
We define the $\text{cartesian product}$ of this family, denoted by $$A_1\times ...\times A_m $$ to be the set of all $m-tuples$ $(x_1,...,x_m)$ of elements of $X$ such that $x_i\in A_i$ for each $i.$
In particular taking $A_i=$ $\Bbb R$ for $i=1,...,m$ we get a rigorous definition of the $\text {cartesian product}$ on $\Bbb R$ denoted by, $$\Bbb R \times...\times \Bbb R \text{[m-times]}=\Bbb R^m$$
A: In math there is a tendency to refer to objects that are technically different, but which are closely related, by the same name. So depending on the context, $\mathbb R^n$ could denote the set of all functions from $\{1,2,\ldots, n\}$ to $\mathbb R$, or it could denote the set of all $n \times 1$ matrices with real entries. And technically, those are two different things. Hopefully the meaning will be clear from context. 
By the way, if you dig into the definitions, a real $n \times 1$ matrix is often defined to be a function from $\{(i,1)\mid 1\leq i \leq n\}$ to $\mathbb R$.
