Because of the fact that $exp:\mathfrak{so}(n)\to SO(n)$ is surjective it follows that $SO(n)$ is connected?

Question

Because of the fact that $exp:\mathfrak{so}(n)\to SO(n)$ is surjective it follows that $SO(n)$ is connected?

I am looking at a proof that $SO(n)$ is connected and in this show that the function $exp:\mathfrak{so}(n)\to SO(n)$ is surjective, because it is only necessary to do this? Thank you.