Representation a matrix as a colored square shape

Update 2020-01-07

At this point there is a Wolfram Function Repository (WFR) RandomScribble.

Here is matrix representation with "midriff" random scribbles based WFR's RandomScribble:

matrix = 
   {{1, -1, 1, 1, -1}, 
    {-1, -1, -1, 1, 1}, 
    {1, 1, 1, -1, 1}, 
    {-1, 1, -1, 1, -1}, 
    {1, 1, 1, -1, -1}};

SeedRandom[23];
Magnify[Grid[
  matrix /. {-1 :> OrangeScribble[], 1 :> DarkGreenScribble[]}, 
  Dividers -> All], 0.4]

(Note that this answer by @kglr has a different approach to making "midriff" scribbles. )

Definitions

Clear[OrangeScribble];
OrangeScribble[opts___] := 
  ResourceFunction["RandomScribble"][opts, 
   "EnvelopeFunctions" -> Automatic, "NumberOfStrokes" -> 200, 
   "RotationAngle" -> Pi/4, ColorFunction -> "SiennaTones", 
   PlotStyle -> {Orange, AbsoluteThickness[3]}];

Clear[DarkGreenScribble];
DarkGreenScribble[opts___] := 
  ResourceFunction["RandomScribble"][opts, 
   "EnvelopeFunctions" -> Automatic, "NumberOfStrokes" -> 200, 
   "RotationAngle" -> Pi/4, ColorFunction -> "AvocadoColors", 
   PlotStyle -> {Orange, AbsoluteThickness[3]}];

---

Original answer

matrix = 
{{1, -1, 1, 1, -1}, 
{-1, -1, -1, 1, 1}, 
{1, 1, 1, -1, 1}, 
{-1, 1, -1, 1, -1}, 
{1, 1, 1, -1, -1}};

SeedRandom[23];
Grid[matrix /. {-1 :> OrangeScribble[], 1 :> DarkGreenScribble[]}, Dividers -> All]

Update

A problem: If I change the number of raws and columns, then the size of the shape goes very big. I wish to change my matrix to for example :10*10.

matrix2 = ArrayFlatten[Table[matrix, 2, 2]];

Magnify[Grid[
  matrix2 /. {-1 :> OrangeScribble[], 1 :> DarkGreenScribble[]}, 
  Dividers -> All], 0.4]

Definitions

Clear[RandomScribble];
Options[RandomScribble] = 
  Join[{AbsoluteThickness -> 2, ColorFunction -> ColorData[87]}, Options[Graphics]];
RandomScribble[nPoints_Integer, opts : OptionsPattern[]] := RandomScribble[nPoints, \[Pi]/4, opts];
RandomScribble[nPoints_Integer, dir_?NumericQ, opts : OptionsPattern[]] :=
  Block[{r, absTh, colorFunc},
   
   absTh = OptionValue[RandomScribble, AbsoluteThickness];
   colorFunc = OptionValue[RandomScribble, ColorFunction];
   
   If[! (NumericQ[absTh] && absTh > 0), Return[$Failed]];
   
   r = RandomReal[{-1, 1}, {nPoints, 2}];
   
   Graphics[{
     AbsoluteThickness[absTh], colorFunc[0], 
     BezierCurve[Sort[r].RotationMatrix[dir]]
     },
    FilterRules[{opts}, Options[Graphics]]]
   ];

OrangeScribble[] := RandomScribble[RandomInteger[{160, 190}], \[Pi]/4, ColorFunction -> (Orange &), ImageSize -> Tiny];
DarkGreenScribble[] := RandomScribble[RandomInteger[{120, 180}], \[Pi]/4, ColorFunction -> (Darker[Green] &), ImageSize -> Tiny];

As @AntonAntonov says, use MatrixPlot:

matrix = {{ 1, -1,  1,  1, -1},
          {-1, -1, -1,  1,  1},
          { 1,  1,  1, -1,  1},
          {-1,  1, -1,  1, -1},
          { 1,  1,  1, -1, -1}};

MatrixPlot[matrix, 
           ColorRules -> {-1 -> Orange, 1 -> Darker@Green},
           Frame -> False]

I misunderstood the meaning of square color before, here is my new answer:

matrix = Array[RandomChoice[{-1, 1}] &, {10, 10}];
positionA = Position[matrix, 1];
positionB = Position[matrix, -1];
Grid[matrix, Frame -> All, 
 Background -> {None, None, 
   Join[Rule[#, Green] & /@ positionA, 
    Rule[#, Orange] & /@ positionB]}, 
 ItemStyle -> {Automatic, Automatic, 
   Join[Rule[#, White] & /@ positionA, 
    Rule[#, Black] & /@ positionB]}]