It is a standard exercise to show that $(GL_n(\mathbb{R}), \tau, D)$ is a smooth manifold, where
$$\tau = \{\phi^{-1} (U) \subseteq GL_n (\mathbb{R}) | U\in \mathbb{R}^{n^2} is \quad open, \phi: GL_n(\mathbb{R}) \to \mathbb{R}^{n^2} \quad \text{s.t $\phi$ sends the entries of a matrix to an $n^2$-tuple with the same entries} \},$$ and $$D = \{f\circ\phi:GL_n (\mathbb{R})\to \mathbb{R}^k | f: V\subseteq \mathbb{R}^{n^2}\to \mathbb{R}^k \text{is a a differentiable map in the usual sense}\}.$$
However, I would like to find a smooth atlas for this smooth manifold $(GL_n(\mathbb{R}), \tau, D)$ explicitly, i.e explicitly smooth coordinate patches that they cover the whole set altogether.