the map exp for compact real lie groups

Question

Something I'm reading says that for a compact connected complex Lie group $G$, the kernel of $\exp:T_e(G) \to G$ is a lattice $\Lambda$ in $T_e(G)$, and $\exp$ is surjective, so $G \simeq T_e(G)/\Lambda$. What is the story for real compact connected Lie groups? Looking at the classification of compact Lie groups, get things besides tori, e.g. $SO(n)$ and $SU(n)$. In these cases, is $\exp$ injective, surjective, or is the kernel a lattice and is the image a maximal torus?