How to plot a distribution of planes

a = .2;
base = Polygon[{{-a, -a, 0}, {-a, a, 0}, {a, a, 0}, {a, -a, 0}}/2];
r = 2 a;

Graphics3D[{

  GeometricTransformation[
   base,
   Flatten[
    Table[
         TranslationTransform[{x, y, 0}] @* RotationTransform[
            {{0, 0, 1}, Cross[{1, 0, .5 y}, {0, 1, -.5 x}]}],
     {x, - r, r, a}, {y, - r, r, a}
     ], 1]
   ],
  {  Thick
     ,
     Blue, (* normal to base*)
     Arrow[{#, # + {0, 0, .2}} ]
     ,
     Green, (* spanning vectors *)
     Arrow[{#, # + .2 {1, 0, .5 #[[2]]}} ], 
     Arrow[{#, # + .2 {0, 1, -.5 #[[1]]}} ]
     ,
     Red, (* normal to spanned plane*)
     Arrow[{#, # + .2 Cross[{1, 0, .5 #[[2]]}, {0, 1, -.5 #[[1]]}]  }]
  } &@{r, r, 0}
 }, 
  PlotRange -> All, Axes -> True]

and with J.M.'s suggestion about BoxRatios:

Graphics3D[{
  base,
  GeometricTransformation[
   base,
   Flatten[
    Table[
     TranslationTransform[{x, y, 0}]@*RotationTransform[
       {{0, 0, 1}, Normalize@Cross[{1, 0, .5 y}, {0, 1, -.5 x}]}],
     {x, -r, r, a}, {y, -r, r, a}
     ], 1]
   ]

  }, PlotRange -> All, Axes -> True, BoxRatios -> {1, 1, 0.4`}]

a = .1;
r = 5 a;

Here's a very compact version, constructed through the magic of dot products:

With[{a = 1/5, r = 2/5}, 
     Graphics3D[Polygon[Flatten[Table[{x, y, 0} + # & /@
                                      (a {{1, 1}, {-1, 1}, {-1, -1}, {1, -1}} .
                                      {{1, 0, y/2}, {0, 1, -x/2}}/2),
                                      {x, -r, r, a}, {y, -r, r, a}], 1]],
                Axes -> True, BoxRatios -> {1, 1, 0.4}, 
                Lighting -> "Classic", PlotRange -> All]]