Why do fabric skin constructions not show up much in larger single engine piston aircraft?
This answer assumes that "GA" means "little piston planes."

Aluminum (or non-fabric) aircraft tend to "hold up" better against the elements.  I would not store a Stinson 108 outdoors but would not think twice about storing a 172 (or Cirrus) on an open air tiedown.  The fabric covering requires a great deal of care compared to the metal (or composite) skin of most GA planes.

Fabric planes must be re-covered every few decades, more frequently if neglected.  This is a non-trivial operation with a price tag to match.  This typically involves reconditioning the now-exposed structure which adds significant cost.  Re-covering and painting a Cub (small plane), for example, can cost an owner tens of thousands of dollars. Stripping, treating, and painting a "metalized" Cessna 120 (similar to a Cub) is a much easier, faster, and cheaper task.

Another consideration is the availability of aluminum.  Aluminum was declared a strategic material in WW2 and, as a result, a great deal of energy was allocated to increase production.  With the war over and the infrastructure in place to produce a lot of aluminum cheaply, why adhere to an antiquated manufacturing process requiring a painstaking level of detail when lower skilled labor can use cheap aluminum to mass produce high quality planes?

In short, fabric planes typically require more care and attention than their stressed skin cousins.  They also require somewhat specialized and skilled labor to build and maintain.  This generally translates into higher costs for everyone involved.

I'm sure there are technical reasons for the shift away from fabric but I'll let Peter (or another technical expert) comment on those.