Suppose that I have a group $G$ that is either $SU(n)$ (special unitary group) or $SO(n)$ (special orthogonal group) for some $n$ that I don't know. Which "questions" should I ask to determine which one it is? e.g. which structural differences are there between these groups? Of course a question like "Is $G$ isomorphic to $SU(n)$ for some $n$?" doesn't count.
I'm hoping for some answer that doesn't refer to the smooth structure (the fact that they are Lie groups), or even to the topology altough that might be asking too much.
edit: to make it a bit clearer what kind of answer I'm looking for: I'm trying to understand better the differences between regular quantum theory and real-valued quantum theory. In the first you can generate all pure states by starting with a pure state and then looking at its orbit under a unitary group, while in the second you have to look at the orbit under the orthogonal group. I'm hoping for some kind of intuitive property that makes $SU(n)$ a more natural choice for a symmetry group then $SO(n)$. A reason why nature has 'chosen' complex quantum theory over real quantum theory.
For $n\ge 3$, the group $SO(n)$ has either trivial center (if $n$ is odd) or center of order 2 (if $n$ is even). In contrast, the center of $SU(n)$ ($n\ge 2$) is isomorphic to the group of $n$th roots of unity and, hence, has order $\ge 3$ whenever $n\ge 3$.
However, the differences between these groups go much deeper than the center. The groups $SO(n), n\ge 5$ and $SU(n), n\ge 3$ have simple Lie algebras of different type: Type A for $SU(n)$ and type B (or D, depending on the parity of $n$) for $SO(n)$. This makes a world of difference when you study these groups (it affects representation theory of these groups among other things). However, in order to appreciate this difference you have to think of these as Lie groups, not abstract groups. Refusing to deal with the Lie-theoretic aspects of these groups is akin to trying to learn probability without measure theory or real analysis without limits. My suggestion is to start learning about roots, weights, etc.