General Linear group over $V$- vector space I have a confusion regarding what kind of objects might be appearing in the entries of  matrices in $\operatorname{GL}(V)$. For a simple example, if I have a finite dimensional $\mathbb{R}$- vector space $\mathbb{R}^n$, then the entries of the matrix will be representing a vector in an $n$-dimensional space or just a real number ? I know  if  I have a $n$-dimensional real vector space, it will need $n$ elements to form a basis such as  $\{e_1,e_2,\dotsc,e_n\}$ , but is $e_i$ a vector with $n$ components? or is it just a real number ? is it possible that it could be a geometrical object or a matrix itself?
My second question is that if i were to choose a basis of this vector space and wanted to use them to identify the group $G=\operatorname{GL}(V)$ and the space of symmetric bilinear forms on this vector space with spaces of matrices , how would I do that?
P.S: $V$ is $\mathbb{R}^n$.
 A: Given a basis $\{v_1, \dots, v_n\}$ of an $\mathbb{R}$-vector space $V$, we can identify $GL(V)$, the set of invertible linear transformations $V \to V$, with $GL_n(\mathbb{R})$, the set of non-singular $n \times n$ matrices with entries in $\mathbb{R}$.
Given $f \in GL(V)$, for each $j$ with $1 \leq j \leq n$, write
$$
f(v_j) = \sum_{i=1}^n a_{ij}\, v_i.
$$
This linear combination will exist since the basis spans, and the weights will be unique since the basis is linearly independent. In other words, we have the map
\begin{align*}
GL(V) \quad &\to \quad GL_n(\mathbb{R}) \\
f \quad &\mapsto \quad (a_{ij}).
\end{align*}
The same formula, together with linearity, allows us to define the inverse map. Given a matrix $(a_{ij}) \in GL_n(\mathbb{R})$, consider an arbitrary $v \in V$. We need to define $f(v) \in V$ so that we obtain $f \in GL(V)$. First express $v$ in terms of the basis:
$$
v = \sum_{j=1}^n c_j\, v_j. 
$$
Now,
\begin{align*}
f(v) &= f\biggl(\sum_{j=1}^n c_j \, v_j \biggr) \\
&= \sum_{j=1}^n c_j \, f(v_j) \\
&= \sum_{j=1}^n c_j \, \sum_{i=1}^n a_{ij} \, v_i \\
&= \sum_{i=1}^n \biggl(\sum_{j=1}^n c_j \, a_{ij}\!\biggr) \, v_i.
\end{align*}
This is pretty clearly linear, but we have to check that this is indeed invertible. Thus, we have
\begin{align*}
GL_n(\mathbb{R}) \quad &\to \quad GL(V) \\
(a_{ij}) \quad &\mapsto \quad f.
\end{align*}
