How is a gameobject rotated when you set it's forward?

I'm doing my own Unity with OpenTK, and want to be able to rotate an object by setting its forward, right or up.

I already did the get of the vectors:

         public Vector3 forward
         {
             get
             {
 
                 return Vector3.TransformVector(Vector3.UnitZ, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }
             
         }
 
 
 
         public Vector3 up
         {
             get
             {
 
                 return Vector3.TransformVector(Vector3.UnitY, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }
 
         }
 
         public Vector3 right
         {
             get
             {
                 
                 return Vector3.TransformVector(Vector3.UnitX, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }

but now I need the sets.

Here is the whole component code. using System; using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks; using OpenTK;

 namespace DemoEngine
 {
     public class Transform : Component
     {
         private float GaR = (float)Math.PI / 180;
         private float RaG = 180 / (float)Math.PI;
         public Vector3 position;
         public Vector3 rotation;
         public Vector3 scale = Vector3.One;
         private List<Transform> childList;
         private Transform _parent;
         public Transform parent
         {
             get
             {
                 return _parent;
             }
             set
             {
                 if (_parent != null)
                 {
                     _parent.childList.Remove(this);
                 }
                 _parent = value;
                 value.childList.Add(this);
             }
         }
 
         public Vector3 forward
         {
             get
             {
 
                 return Vector3.TransformVector(Vector3.UnitZ, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }
             
         }
 
 
 
         public Vector3 up
         {
             get
             {
 
                 return Vector3.TransformVector(Vector3.UnitY, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }
 
         }
 
         public Vector3 right
         {
             get
             {
                 
                 return Vector3.TransformVector(Vector3.UnitX, Matrix4.CreateRotationX(rotation.X) * Matrix4.CreateRotationY(rotation.Y) * Matrix4.CreateRotationZ(rotation.Z));
             }
 
         }
         public Transform()
         {
             childList = new List<Transform>();
         }
 
         public Matrix4 GenerateMatrix()
         {
 
             Matrix4 matrix = Matrix4.Scale(scale) *
                                 Matrix4.CreateRotationX(rotation.X) *
                                 Matrix4.CreateRotationY(rotation.Y) *
                                 Matrix4.CreateRotationZ(rotation.Z) *
                                 Matrix4.CreateTranslation(position);
 
             if (_parent != null)
             {
                 matrix *= _parent.GenerateMatrix();
             }
 
             return matrix;
         }
 
         public float xRotation
         {
             get
             {
                 return rotation.X * RaG;
             }
             set
             {
                 rotation.X = value * GaR;
             }
         }
         public float yRotation
         {
             get
             {
                 return rotation.Y * RaG;
             }
             set
             {
                 rotation.Y = value * GaR;
             }
         }
         public float zRotation
         {
             get
             {
                 return rotation.Z * RaG;
             }
             set
             {
                 rotation.Z = value * GaR;
             }
         }
 
     }
 
 
 }