Here $GL_n$ is the set of all invertible $n\times n$ real matrices with some other structure. Regard $\mathbb R^{n\times n}$, the set of all $n\times n$ matrices, as an $n^2$ dimensional real vector space, a smooth manifold with standard differential structure. Consider the determinant function $\det_n:\mathbb R^{n\times n}\to \mathbb R$, it's a smooth function, also it's continuous. $\mathbb R-\{0\}$ is an open set in $\mathbb R$, so $GL_n=\det_n^{-1}(\mathbb R)$ is an open submanifold of $\mathbb R^{n\times n}$. It's also a group with matrix multiplication, in which multiplication and inverse maps are smooth, so it's a Lie group.
And $gl_n$ is the set equal to $\mathbb R^{n\times n}$ with some other structure. As I've shown above, it has a vector space structure. Matrix multiplication makes it an associative algebra, which concludes a Lie bracket.
As I've learnt, Lie groups can acts on itself by multiplycation. For a Lie group $G$, note $l_g:G\to G,p\mapsto gp$, considered as a left action for each $g\in G$. A globle tangent vector fields $X$ satisfying $(l_g)_*X_p=X_{gp}$ for each $g,p\in G$ is called a left invariant tangent vector field. All left invariant tangent vector fields can become into a Lie subalgebra of $\Gamma^\infty(G,TG)$, noted as $L(G)$. Because $X_p=(l_p)_*X_1$, where $1$ is the indentity of $G$, it's easy to show that vactor space $L(G)\cong T_1G\cong \mathbb R^n$, where $n$ is the dimension of $G$. So computing Lie algebras is computing Lie brackets.
It's easy to show that $L(\mathbb R^n)\cong\mathbb R^n$ with $[X,Y]=0$, and $L(S^1)\cong \mathbb R$ with $[X,Y]=0$. It's hard to compute other examples. I've computed that $L(GL_n)\cong gl_n$ by a not-so-strict way. To do this, I used 2 theorems, which has been shown strictly.
\textbf{Thm.1.} For an $m$ dimensional smooth manifold $M$ and its coordinate card $\langle U,\varphi=(x_1,\cdots,x_m)\rangle$, and an $n$ dimensional smooth manifold $N$ and its coordinate card $\langle V,\psi=(y_1,\cdots,y_m)\rangle$, and a smooth germ $g$ of $N$ at $F(p)$, and a map $F$ smooth at $p\in U$ satisfying $F(p)\in V$, the tangent map $F_*:T_pM\to T_{F(p)}N$ satisfys that
$F_*\frac{\partial}{\partial x_j}\bigg|_pg =\frac{\partial}{\partial x_j}\bigg|_p(g\circ F) =\frac{\partial\left(g\circ F\circ\varphi^{-1}\ \right)} {\partial x_j}\bigg|_p$
$=\displaystyle\sum_i \frac{\partial\left(g\circ\psi^{-1}\ \right)}{\partial y_i}\bigg|_{F(p)} \frac{\partial\left(y_i\circ F\circ\psi^{-1}\ \right)} {\partial x_j}\bigg|_p =\displaystyle\sum_i \frac{\partial\left(y_i\circ F\circ\psi^{-1}\ \right)} {\partial x_j}\bigg|_p \frac{\partial}{\partial y_i}\bigg|_{F(p)}g$
So the matrix of $F_*$ is the Jacobian matrix, with basis $\left\{\frac{\partial}{\partial x_j}\bigg|_p\right\}_j$ and $\left\{\frac{\partial}{\partial y_i}\bigg|_{F(p)}\right\}_i$.
\textbf{Thm.2.} For For an $m$ dimensional smooth manifold $M$ and its coordinate card $\langle U,\varphi=(x_1,\cdots,x_m)\rangle$, I note $\partial_k:=\frac{\partial}{\partial x_k}$. For $X=\displaystyle\sum_k a_k\partial_k, Y=\displaystyle\sum_l b_l\partial_l$, and $p\in U$, $f$ a smooth germ at $p$,
$[X,Y]_pf=\displaystyle\sum_{k,l} a_k(p)\partial_k|_p(b_l\partial_lf)-b_l(p)\partial_l|_p(a_k\partial_kf)$
$=\displaystyle\sum_{k,l}\left (a_k(p)(\partial_k|_pb_l)\partial_l|_pf) +a_k(p)b_l(p)\partial_k|_p\partial_lf -b_l(p)(\partial_l|_pa_k)\partial_k|_pf -b_l(p)a_k(p)\partial_l|_p\partial_kf)\right)$
$=\displaystyle\sum_{k,l}( a_k(\partial_kb_l)\partial_l-b_l(\partial_la_k)\partial_k)_pf$
The 2 theorems above are proved strictly, but the computing below is not. Here's my computing.
$GL_n$ is an open submanifold of $\mathbb R^{n\times n}$. Note $\partial_{i,j}:=\frac{\partial}{\partial x_{i,j}}$. For a matrix $M\in\mathbb R^{n\times n}$, note $M_{i,j}$ as the element in the i-th row, the j-th column, note $M=(M_{i,j})_{i,j}$. For $M,p\in GL_n$, $\frac{\partial(Mp)_{i,j}}{\partial p_{k,l}} =\displaystyle\sum_r\frac{\partial M_{i,r}p_r,j}{\partial p_{k,l}} =\delta_l^jM_{i,k}$, so $(l_M)_*\partial_{k,l}|_p =\displaystyle\sum_iM_{i,k}\partial_{i,l}|_{Mp}$.
For $X=\displaystyle\sum_{k,l}x_{k,l}\partial_{k,l} \in\Gamma^\infty(GL_n,TGL_n)$, and $M,p\in GL_n$, $(l_M)_*X_p =\displaystyle\sum_{k,l}x_{k,l}(p)\sum_iM_{i,k}\partial_{i,l}|_{Mp} =\displaystyle\sum_{i,l} \left(\displaystyle\sum_kM_{i,k}x_{k,l}(p)\right)\partial_{i,l}|{Mp}$. For $v=\sum_{i,j}a_{i,j}\partial_{i,j}|_p$, note $[v]=(a_{i,j})_{i,j}$, then $[(l_M)_*X_p]=M[X_p]$. $X\in L(GL_n)$ iff $[X_{Mp}]=M[X_p]$. Then $[X_p]=p[X_I]$, where $I$ is the identity metrix.
Suppose $X,Y\in GL_n$ that $[X_I]=(a_{i,j})_{i,j}=:A,[Y_I]=(b_{k,l})_{k,l}=:B$, then $\partial_{i,j}[X]_{k,l} =\displaystyle\sum_r \frac{\partial p_{i,r}a_{r,j}}{\partial p_{k,l}} =\delta_k^ia_{l,j},\partial_{k,l}[Y]_{i,j} =\displaystyle\sum_s \frac{\partial p_{k,s}b_{s,l}}{\partial p_{i,j}} =\delta_i^kb_{j,l}$. Then $[X,Y]_p =\displaystyle\sum_{i,j,r,l}p_{i,r}a_{r,j}b_{j,l}\partial_{i,l} -\displaystyle\sum_{k,l,s,j}p_{k,s}b_{s,l}a_{l,j}\partial_{k,j}$, that is, $[[X,Y]_p]=p(AB-BA)$. Thus $L(GL_n)\cong gl_n$.
My computing is shown above, but it's not so strict, because I use tangent vector fields of $\mathbb R^{n\times n}$ but not $GL_n$. The answer is right, but the method is not strict. And this method cannot compute $L(SL_n)$ and stuff. Could you please give me a more strict, more general way to compute Lie algebras of Lie groups?