Understand the suspension (topology) and some Lie group examples Let $\Sigma$ denotes a suspension 
$$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$ 
where $\wedge$ and $\vee$ are the smash product and the wedge sum (one point union) of pointed topological spaces respectively.
I understand the wedge sum is a "one-point union" of a family of topological spaces. But the suspension is harder to imagine.
(1) So what are the examples that we can identify the suspension $\Sigma X$ for
(special) unitary groups and (special) orthogonal groups?


*

*$\Sigma X = \Sigma SU(n)=?$

*$\Sigma X = \Sigma U(n)=?$

*$\Sigma X = \Sigma SO(n)=?$

*$\Sigma X = \Sigma O(n)=?$
(2) What is the good way to understand intuitively the suspension (topology) in terms of simple pictures?
 A: As Randall said in his comment, suspensions of "well-known spaces" in general do not produce something common. But you can imagine how it looks like via the picture in your question The suspension (topology) and elementary examples  - provided you can imagine how spaces like $SU(n)$ look like ;-)
As far as I know, the suspensions in your examples are no spaces denoted by a "standard name".
I guess your interest in suspensions comes from algebraic topology, at least I do not know many other places where there are relevant. In this context it suffices to consider homotopy types of spaces. As I mentioned in my answer to your other question, you may sometimes find something "more common" homotopy equivalent to a suspension.
Moreover, for any pointed space $(X,x_0)$ with a non-degenerate basepoint the reduced suspension $\Sigma (X,x_0)$ has the same homotopy type as the unreduced suspension $S X$ (which is easier to imagine). Examples for such spaces are all manifolds, CW-complexes and polyhedra.
Finally it is remarkable that sometimes suspensions of "exotic spaces" yield well-known spaces (that is, you get a sort of improvement). See for example https://en.wikipedia.org/wiki/Double_suspension_theorem.
