I'm trying to construct an atlas the painful way for $SU(n)$, using charts. According to Wiki, SU(n) is a real manifold of real dimension $n^2-1$; I can believe this because you can perform a Cayley Transformation for any $U \in SU(n)$ such that the map $C:M_n(\mathbb{C})^{*}\longrightarrow M_n(\mathbb{C})$
$$ C : U \longmapsto C(U) = (1_n - U)(1_n+U)^{-1} $$
Which is well defined because any unimodular unitary matrices are non exceptional (that is that have $\det(1_n+U)\neq 0$). Thus the Cayley transformation defines a homeomorphism, given that you can use the Lie algebra
$$\mathfrak{su}(n) = \{X \in GL_n(\mathbb{C}): X^{\dagger}+X=0,\; Tr(X)=0\} $$
to explicitly compute the dimension.
But what I want to try is to find explicit charts for these guys. This is what I've tried so far (restricting to $SU(3)$ to see the machinery involved); given that
$$SU(3) = \left\{\;\;\left(\begin{smallmatrix}a&b& c\\ d & e & f\\ g & h & i\end{smallmatrix}\right) \; \left| \right. \left(\begin{smallmatrix}a&b& c\\ d & e & f\\ g & h & i\end{smallmatrix}\right)^{\dagger}\left(\begin{smallmatrix}a&b& c\\ d & e & f\\ g & h & i\end{smallmatrix}\right)=\left(\begin{smallmatrix}1&0&0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{smallmatrix}\right) \; \& \;|ei-hf|^2+|gf-di|^2+|dh-ge|^2 +|dh-ge|^2=1 \;\;\right\} $$
Where you get a whole bunch of conditions from the unitary conditions, and thus a whole set of equivalent conditions for the determinant. I got these by setting the inverse equal to the hermitian conjugate:
Unitary Conditions \begin{eqnarray} a=\overline{ei-hf}\\ b=\overline{gf-di}\\ c=\overline{dh-ge}\\ d=\overline{ch-bi}\\ e=\overline{ai-gc}\\ f=\overline{bg-ah}\\ g=\overline{bf-ce}\\ h=\overline{dc-af}\\ i=\overline{ae-bd} \end{eqnarray}
Determinant Examples \begin{eqnarray} a(ei-hf)-b(di-gf)+c(dh-ge)=1\\ |ei-hf|^2+|gf-di|^2+|dh-ge|^2=1\\ |a|^2+|b|^2+|c|^2=1 \;\;\; \text{etc.} \end{eqnarray}
Now in trying to construct a chart, going by the fact $a,\ldots,i \in \mathbb{C}$, there are 18 real degrees of freedom/9 complex. Since $\mathbb{C}^4 \cong \mathbb{R}^8$, I think it may be easier to work with $\mathbb{C}^4$, since you can always express the matrix entries explicitly using four entries.
And then this is where I think it best to write out a chart, say for $U_a :=\{ei-hf\neq 0 \Leftrightarrow a\neq 0\}$, which is open, as
$$\phi_a : U_a \longrightarrow \phi_a(U_a) \subseteq GL_2(\mathbb{C}) \subseteq \mathbb{C}^4
$$
$$\phi_a (\left(\begin{smallmatrix}a&b& c\\ d & e & f\\ g & h & i\end{smallmatrix}\right)) = (e,f,h,i) \;\;\;\text{equivalently} \left(\begin{smallmatrix}e & f\\h& i\end{smallmatrix}\right)
$$
Then, trying to invert it, things go a bit awry, and when I try to recalculate the explicit inverse, my construction seems to fail, since everytime I try to include the unitary condition and determinant condition in general, things go funny.
Any ideas would be really appreciated!