SU(2)
I know we can view the group element in the SU(2) Lie group as $$ g = \exp\left(\theta\sum_{k=1}^{3} i t_k \frac{\sigma_k}{2}\right) $$ where $(t_1,t_2,t_3)$ forms a unit vector [effectively pointing in some direction on a unit 2-sphere $S^2$], and $\sigma_k$ are Pauli matrices: \begin{align} \sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \,. \end{align} Notice that any group element on $SU(2)$ can be parametrized by some $\theta$ and $(t_1,t_2,t_3)$. Also $\theta$ has a periodicity $[0,4 \pi)$, instead of $2 \pi$. Notice that in this case we also have $$ g = \exp\left(\theta\sum_{k=1}^{3} i t_k \frac{\sigma_k}{2}\right) =\cos(\frac{\theta}{2})+i \sum_{k=1}^{3} t_k \sigma_k\sin(\frac{\theta}{2})$$
question 1: SU(3)
(1) Is this true that all $SU(3)$ group elements can be written as: $$ g = \exp\left(\theta\sum_{k=1}^{8} i t_k \frac{\lambda_k}{2}\right)=\cos(\frac{\theta}{2})+i \sum_{k=1}^{8} t_k \lambda_k\sin(\frac{\theta}{2}) $$ where $\lambda_k$ are Gell-Mann_matrices? And does the second equality still hold? Here Tr$(\lambda_k^2)=2.$ Also $\theta$ has a periodicity $[0,4 \pi)$?
question 2: SU(n), for $n=4, ...$
(2) Is this true that all $SU(4)$ group elements can be written as: $$ g = \exp\left(\theta\sum_{k=1}^{4^2-1} i t_k \frac{\lambda_k}{2}\right)=\cos(\frac{\theta}{2})+i \sum_{k=1}^{4^2-1} t_k \lambda_k\sin(\frac{\theta}{2}) $$ where $\lambda_k$ are generalized rank-4 Gell-Mann_matrices in eq.(3)? And does the second equality still hold? Here Tr$(\lambda_k^2)=2.$ Also $\theta$ has a periodicity $[0,4 \pi)$?
(3) How to determine the Right Hand side equation and $\theta$ periodicity?
There are two key facts that contribute the SU(2) formula for Pauli matrices:
The importance can be seen by squaring a "vector" with them as basis:
$$ (it_1\sigma_1+it_2\sigma_2+it_3\sigma_3)^2=-(t_1^2+t_2^2+t_3^2) $$
(working this algebra out is instructive). This means $t:=it_1\sigma_1+it_2\sigma_2+it_3\sigma_3$ is a square root of $-I$ when $(t_1,t_2,t_3)$ is a unit vector, and thus by the usual proof of de Moivre's formula,
$$ \exp(\theta t)=\cos(\theta)I+\sin(\theta)t. $$
However the Gell-Mann matrices are not as nice. First notice
$$ (i\lambda_1)^2=(i\lambda_2)^2=\mathrm{diag}(-1,-1,0) \\ (i\lambda_4)^2=(i\lambda_5)^2=\mathrm{diag}(-1,0,-1) \\ (i\lambda_6)^2=(i\lambda_7)^2=\mathrm{diag}(0,-1,-1) $$
so these are not quite square roots of $-I$, although that could potentially be fixed in a tentative exponential formula since their sum is a multiple of $-I$. But then the symmetry gets broken:
$$ (i\lambda_3)^2 = \mathrm{diag}(-1,-1,0) \\ (i\lambda_8)^2=\mathrm{diag}(-\frac{1}{3},-\frac{1}{3},-\frac{4}{3}). $$
Thus, in particular,
$$\exp(\theta i\lambda_8)\ne \cos(\theta)+i \sin(\theta)\lambda_8.$$
Moreover, one can verify some of the pairs of Gell-Mann matrices do not anticommute.