Can we parameterise $SU(3)$ in such a way that there are clearly 2 parameters corresponding to the cartan torus? We can parameterise the lie algebra of $SU(3)$ using the Gell-Mann matrices, so that a general element of LA is $\theta_i T_i$, where $T_i=\lambda_i/2$ and $\lambda_i$ are the Gell-Mann matrices. 
Then, since $SU(3)$ is compact and (simply) connected, we can express any element of it in the form $e^{\theta_i T_i}$. However, actually computing this exponential is rather difficult, since it is just a generic traceless hermitian matrix.
The nice thing about this parameterisation is that two of the matrices in the basis generate the Cartan Subalgebra of $\mathcal{L}(SU(3))$, and so it is clear which of the coordinates $(\theta_i)$ correspond to the toroidal subgroup of $SU(3)$.
Is there some way to explicitly parameterise $SU(3)$ (and not just the Lie algebra) so that it is clear which parameters/coordinates correspond to the cartan torus? Essentially, the problem with the above parameterisation is that I can't explicitly compute the exponential, and therefore I can't get an explicit form of the $SU(3)$ matrices.
 A: The group manifold $SU(3)$ can be parametrized almost everywhere according to:
$$ SU(3) \ni g = v(u, \bar{u}) e^{i \theta_3 \lambda_3 + i \theta_8 \lambda_8}$$
where   $v(u, \bar{u})$ is a unitary matrix depending on a three dimensional complex vector $u = (u_1, u_2, u_3)$. This matrix parametrizes the flag manifold $SU(3)/(U(1)\times U(1))$ (which is therefore 6-dimensional). It is given for example in: Daoud and Jellal's work (equation (23)), included in this answer for completeness:
$$v(u, \bar{u}) = \begin{pmatrix}
\frac{1}{\sqrt{\Delta_1}} &  -\frac{\bar{u_1}+u_2\bar{u_3}}{\sqrt{\Delta_1\Delta_2}} & -\frac{\bar{u_3}-\bar{u_1}\bar{u_2}}{\sqrt{\Delta_2}} \\ 
\frac{u_1}{\sqrt{\Delta_1}}  &  \frac{1+|u_3|^2 - u_1u_2\bar{u_3}}{\sqrt{\Delta_1\Delta_2}} & -\frac{\bar{u_2}}{\sqrt{\Delta_2}} \\ 
\frac{u_3}{\sqrt{\Delta_1}}& \frac{u_2+u_2|u_1|^2 - u_3\bar{u_1}}{\sqrt{\Delta_1\Delta_2}} & \frac{1}{\sqrt{\Delta_2}}
\end{pmatrix}$$
with:
$$ \Delta_1(u, \bar{u}) = 1 + |u_1|^2+|u_3|^2$$
$$ \Delta_2(u, \bar{u}) = 1 + |u_2|^2+|u_3-u_1u_2|^2$$
