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There may be native SketchUp plugins that already do this, but if you specifically wanted to use the OpenStudio API you could write a user script in SketchUp, or a measure that could be used outside of SketchUp with this code in it.

model.getPlanarSurfaces.each do |surface|
  puts "#{surface.name},#{surface.netArea}"
end

model.getPlanarSurfaces gets all surfaces in the model (base surfaces, fenestration, shading, interior partition). And using .netArea instead of .grossArea deducts the area of any sub-surfaces from its parent base surface. You could get other information like surface type, construction, etc. If you do this in SketchUp you could just copy the text out of the ruby console into Excel, or if you want to you can have the script/measure write to a CSV file. There is an example measure on the BCL that Exports Schedules to CSV you could use as a guide.

If you only care about exterior surfaces, there are a few other options. The standard OpenStudio report lists total area of exterior surfaces by construction, and also a count of those surfaces.

The standard EnergyPlus report goes further to list all exterior surfaces individually.