Way to rotate Camera with standart Vectors(forward,right, up)

Hello everyone ,Is there any way to rotate camera using right up and forward vectors?

I usually set forward vector and correct other vector using Cross product but it seems nothing changes even i set something to forward, up and right vectors.

 public void SetOrientation(Vector3 lookat, Vector3 up, Vector3 pos) 
   {
         Vector3 forward = (pos -  lookat).normalized;
         Vector3 right =  (Vector3.Cross(new Vector3(0.0f,1.0f,0.0f),forward  ) ).normalized;
         Vector3 up = Vector3.Cross(forward, right).normalized;
 
         this.transform.forward = forward;
         this.transform.right =  right;
         this.transform.up = up;
 
        this.transform.position = pos
  
 }

Best Regards.