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There's an asymmetry because members of O(n) include rotation with random angle and flips with random axis, for small $n$, the fraction of flips is large, which create peaks at $-1$ and $1$ peaks. Restricting to $SO(n)$ removes the peaks at $-1$, and $1$ remaining density follows cosine formula from Girko

negateFirstRow[A_] := {-A[[1, All]]}~Join~A[[2 ;;, All]];
sampleO[n_] := RandomVariate[CircularRealMatrixDistribution[n]];
sampleSO[n_] := 
  With[{mat = sampleO@n}, If[Det[mat] > 0, mat, negateFirstRow@mat]];

n = 4;
numSamples = 100000;
label = StringForm["Eigenvalues of O(``)", n];
angles = Arg /@ Flatten[Table[Eigenvalues@sampleO@n, {numSamples}]];
Histogram[angles, Automatic, PDF, PlotLabel -> label, 
 AxesLabel -> {"arg", "density"}]

label = StringForm["Eigenvalues of SO(``)", n];
angles = Arg /@ Flatten[Table[Eigenvalues@sampleSO@n, {numSamples}]];
observedPlot = 
  Histogram[angles, {Pi/15}, PDF, PlotLabel -> label, 
   AxesLabel -> {"arg", "density"}];
predictedPlot = 
  Plot[(Cos[2 t] + 2)/(4 Pi), {t, -Pi, Pi}, 
   PlotRange -> {0, 3/(4 Pi)}, 
   PlotLegends -> {TraditionalForm[(Cos[2 t] + 2)/(4 Pi)]}];
Show[observedPlot, predictedPlot]