How can I hide HDRI skybox view from my game view but still get its lighting features in realtime?

Let's say I used a HDRI skybox for my game scene, then I baked my lightmaps. If I build my game like this, a weird skybox will be there, right? Instead, I want to put another skybox like a sunny day for my game view. But, I want to utilize the lighting features of HDRI skybox in realtime as well. At the same time, the other skybox will be there just for view and have no effect on lighting. I'm really confused at this point.

Thanks for your answers in advance!