Firebase data not saving

I am new to firebase and am trying to save the user object (users email) to the database through the .then() on the firebase.auth().createUserWithEmailAndPassword(email, password)

When I try to do this it throws an error:

FIREBASE WARNING: set at /users/-KlMikr6xZnfYC6dFJ6L failed: permission_denied

Basically saying I don't have permission to set the user data to the user node.

I know the function works when, because when I set the security write rules to true the userobj is created in the database.

Not sure what I am doing wrong.

controller.js

angular.module('app')
  .controller('appCtrl', function($scope){

    //empty user object to store info
    $scope.user = {};

    //empty object to accept user password
    $scope.password = {};

    // add login event
    $scope.signUp = function(){
      const email = $scope.user.email;
      const password = $scope.password.setup;

      // create new firebase user
      firebase.auth().createUserWithEmailAndPassword(email, password)
        // add user object to database
        .then(function(){
          var userObj = $scope.user;
          console.log($scope.user);

          // set up database references
          let database = firebase.database();
          let ref = database.ref();
          let usersRef = ref.child('users');

          // push user object to database
          usersRef.push(userObj);

        })
        .catch(function(error) {
          // Handle Errors here.
          var errorCode = error.code;
          var errorMessage = error.message;
          console.log(errorCode);
          console.log(errorMessage);
        });   
    };

    firebase.auth().onAuthStateChanged(function(user) {
      if (user) {
        // User is signed in.
        console.log('user signed in');
        var uid = user.uid;
        console.log(uid);
      } else {
        // No user is signed in.
        console.log('user not signed in');
      }
    });

  })

Firebase database rules

"rules": {
  "users": {
    "$uid": {
        ".read": true,
        ".write": "(auth.uid === $uid)"
    }
  }
}