Let us assume we have the following definition of a tangent space:
Definition of smooth path
Let $X\subset\mathbb{R}^n$. Let $I$ be a real interval. \begin{equation} P \text{ is a smooth path in } X \quad:\Leftrightarrow\quad P:I\rightarrow X \text{ is a differentiable function}~. \end{equation}
Definition: Tangent vector at the identity
Let $P$ be a smooth path with $P(0) = \mathtt{I}$. \begin{equation} t \text{ is the tangent vector of } P \text{ at the idenity} \quad:\Leftrightarrow\quad t=\frac{\partial}{\partial x}P(x)|_{x=0} \end{equation} Moreover, we call $t$ a tangent vector of a space $X$, if a path $P:I\rightarrow X$ exists such that $t$ is tangent vector of a $P$.
Definition: Tangent space
We call the space of all tangent vectors (at the identity), the tangent space (at the identity).
Using these definitions, (how) can we show that the tangent space of all invertible matrices $GL(n)$ is indeed the space of all $n\times n$ matrices?
(I'd like to rely mainly on these definitions and do not use the notion of the exponential map if possible...)