I could be wrong but it seems like what you are looking for is a quaternion. You can't represent any rotated vector by only rotating it around the Z-Axis. A quaternion is simply a vector in space that can represent your rotation and then an amount of rotation around that axis. But constraining the rotation to only rotating around the Z-Axis will limit your degrees of freedom. Hopefully this is what you were trying to explain. If not, correct me and hopefully I can help.

When using column vector notation, the Euler rotation is formulated as
R[sub]z[/sub] * R[sub]y[/sub] * R[sub]x[/sub] =: R[sub]E[/sub]
which you want to equal a single z axis rotation, so that
R[sub]E[/sub] = R[sub]z[/sub]'

This implies 9 equations for a single variable (namely the argument angle to R[sub]z[/sub]'), what obviously can be solved in special situations only. When such a situation is actually given, then you have 4 equations like
cos( z' ) = a
cos( z' ) = b
sin( z' ) = c
-sin( z' ) = d
where the numerical values of a and b as well as those of c and -d are obviously the same; further a and c need to differ in a phase shift of 90 degree.

When you use the arccos or arcsin function for solving, then you get a result that covers a half circle only. So you need to take 2 of the above functions into account. One way to do this is to choose e.g.
tan( z' ) = sin( z' ) / cos( z' ) = a / c
and hence
z' = arctan( a / c )
in principle. However, this formula still has problems, because it (1.) is vulnerable against c==0 and (2.) only 2 of the 4 quadrants can be identified (i.e. the same problem as those above).

Fortunately, most programming languages provide a variant of arctan that internally deals with this problem. This function is usually named atan2. So try
z' = atan2( c, a )
where a and c are picked from the combined Euler rotation matrix.


FYI: The above does something like looking at the problem in the 2D space given by the x/y plane. It sees a vector (a c)[sup]t[/sup] and computes the angle that is between the principal x axis and the vector. You can find several discussions of this in this forum if you search for e.g. "atan2".