The study of student engagement has attracted growing interests to address problems such as low academic performance, disaffection, and high dropout rates. Existing approaches to measuring student engagement typically rely on survey-based instruments. While effective, those approaches are time-consuming and labour-intensive. Meanwhile, both the response rate and quality of the survey are usually poor. As an alternative, in this paper, we investigate whether we can infer and predict engagement at multiple dimensions, just using sensors. We hypothesize that multidimensional student engagement level can be translated into physiological responses and activity changes during the class, and also be affected by the environmental changes. Therefore, we aim to explore the following questions: Can we measure the multiple dimensions of high school student's learning engagement including emotional, behavioural and cognitive engagement with sensing data in the wild? Can we derive the activity, physiological, and environmental factors contributing to the different dimensions of student learning engagement? If yes, which sensors are the most useful in differentiating each dimension of the engagement? Then, we conduct an in-situ study in a high school from 23 students and 6 teachers in 144 classes over 11 courses for 4 weeks. We present the n-Gage, a student engagement sensing system using a combination of sensors from wearables and environments to automatically detect student in-class multidimensional learning engagement. Extensive experiment results show that n-Gage can accurately predict multidimensional student engagement in real-world scenarios with an average mean absolute error (MAE) of 0.788 and root mean square error (RMSE) of 0.975 using all the sensors. We also show a set of interesting findings of how different factors (e.g., combinations of sensors, school subjects, CO2 level) affect each dimension of the student learning engagement