Is there any spin group $Spin(n)$ for any value of $n$ which is isomorphic to $SU(3)$?

Question

I was wondering about the spin group and its relations with the unitary groups. Like we have $Spin(2)\simeq U(1)$, $Spin(3) \simeq SU(2)$.

I was wondering whether this relation can be taken further to $SU(3)$.

The motivation behind this, is that, the unitary group in these cases are all important in physics.

Answer 1

The dimension of $SU(n)$ is $n^2-1$. While $\dim(Spin(m))=\dim(SO(m))=m(m-1)/2$. So we get $n^2-1=m(m-1)/2$.

Substituting $n=3$, we get $8=m(m-1)/2$

Solving we get $m=(1\pm \sqrt{65})/2\not\in\mathbb Q$

A contradiction.