Camera rotation

I'm trying to make a first person camera. Right now I have the camera copy position and rotation of Character Controller component. The body rotates like that:

 void HandleRotation () {

     rotateVector = Vector3.zero;

     rotateVector.y = Input.GetAxisRaw ("Mouse X");

     rotateVector.Normalize ();

     rotateVector.y *= mouseSensitivity;

     characterController.transform.Rotate (rotateVector);

 }

And i have the camera follow its position and rotation:

 void UpdatePosition () {
     cameraObject.transform.position = PlayerLookAt.transform.position;

     cameraObject.transform.rotation = PlayerLookAt.transform.rotation;
 }

The problem is that I want the camera to be able to rotate up so the player can look up and down. I've tried to let the body handle that but due to Character Controllers lack of rotation constraints the whole body rotates. I've also tried to let the camera itself take care of rotation, like this :

 void UpdatePosition () {
     cameraObject.transform.position = PlayerLookAt.transform.position;

     rotateVector = Vector3.zero;

     rotateVector.x = Input.GetAxisRaw ("Mouse Y");

     rotateVector.Normalize ();

     rotateVector.x *= mouseSensitivity;

     cameraObject.transform.Rotate (rotateVector);

     cameraObject.transform.rotation = PlayerLookAt.transform.rotation;
 }

but it gets overridden by rotation of the body. I thought about copying only the "y" rotation from the body and leaving the rest as is but I am not sure how to do that. I cannot change the transform.rotation values by themselves, and I've tried transform.rotation.eulerAngles.Set(), but it doesn't work either.