Problem with transparent shader

I'm testing a character I made and there seems to be something going on between the shader I'm using for the hair and the camera. (shader: Legacy Shaders/Transparent/Cutout/Soft Edge Unlit)

It looks fine when the background behind the hair isn't the skybox however, when that happens, it gets transparent areas where it should be opaque. It displays correctly in Scene view on both situations. Also, when I switch to Don't clear or Depth only in Camera->Clear Flags the problem is solved, but I want to stick with the Skybox option.

I've recently started to use Unity, so my knowledge on the topic is relatively limited. If additional information is needed, let me know. Thanks!