Here are some examples of 3 dimensional transforms and their corresponding 3x3 matrices.
I have noted whether the matrix is symmetric or antisymmetric across the leading diagonal. As you can see, rotation contains the sum of both a symmetric and an antisymmetric across the leading diagonal, reflection is symmetric across the leading diagonal.

Identity (symmetric) 


Scale (symmetric) 


Swap x and y axes, which is the same as reflecting in a 45° line (symmetric) 


Swap x and y axes and invert y, which is the same as rotating by 90° around z axis (antisymmetric) 


Rotate by a around z axis (antisymmetric) 


Reflection in z axis  

Rotation around an axis x,y,z For examples of 3D rotations see this page. 


Skew symmetric matrix, Matrix equivalent of vector cross multiplication, this transform generates a vector which is mutually perpendicular to both x,y,z and the input vector.  

Reflection in line x,y,z (see this page) (symmetric) 
