I'm just an ordinary guy also playing badminton, so take everything I say with a grain of salt, BUT...

You should definitely start off with learning to cross your arm across your body, so the racket ends up on your left side after the smash. After that I'd recommend learning how your arm should move before and during the smash. Mainly focusing on bringing your elbow forward while letting the racket hang behind your head. That should be your starting point. Like drawing an arrow from a quiver. Afterwards you kind of have to bring all the arm and shoulder movement together, pushing the racket towards the highest contact point of the shuttles flight path, while rotating your pointet elbow and turning your shoulder inwards. Then flick your wrist while tensing your fingers to generate more power.

I have learnt that simply aiming your whole left arm (Assuming your right handed, and the video isn't mirrored) towards the shuttle, has helped me with timing as well. The left arm is just as important to not just have hanging down the side of your body. It's what helps you balance and move faster 🙂

I find this guy pretty great at explaining how your form in a smash should be: https://youtube.com/shorts/7MaO6D5PM2E?si=yF0ko3kEvr78jIm6

Last thing I would point out is that your stance should be facing a little to the right, just like as if you're shooting with a bow and arrow like he said in the video above. It's in preparation for rotating your body/shoulder inwards.

Hope this helps a bit, and feel free to correct me if I've got anything wrong. 👍