First, make sure you're not applying lighting to the skybox. It should be full brightness without any lights. Lights make box edges visible even without textures, so you're probably not doing that.
Second, your choice of texture needs to be seamless and well chosen as a skybox texture(s). If you're rendering a literal sky box then the textures need to also be created with knowledge that they're mapped to a box and be distorted in such a way that it appears non-boxy near the seams, essentially the same as a cubemap.
There are more advanced ways of rendering skyboxes which give even better results, but that's the gist of the basic version. Here's a fixed-function OpenGL tutorial that will maybe explain things better than I did.