Skylights are pleasing because they fill the room with light which is reflected and diffused off all the surfaces in the room, whereas a typical ceiling lamp is somewhat unidirectional and lights the ceiling almost as much (or possibly more) than the room itself.
Uplighting from multiple sources is definitely a good solution for recreating the skylight effect. These could be floor mounted, or sconces hung on or cut into the ball. Uplighting on top of cabinets will cast light on the ceiling. You could also couple the uplighting with track-mounted wall wash bulbs.
Halogen lighting was often used for this sort of lighting but LEDs are becoming a better alternative as the prices drop. Good quality LED lighting strips and bulbs are much more low-heat and energy efficient than any halogen. In addition, most LEDs will last at least 10 years without replacement so they can be installed in areas that are hard to access. Traditionally, people consider the traditional yellow tungsten-bulb colors to be most comfortable, which is around 3000K (warm white), but this is not what you want to simulate sunlight. Actual sunlight is around 5000K and above and more of a bluish color (cool white). You will need LEDs around that color temp, or can be programmed to that. Approx 4000K can be a nice compromise. Ultimately everything depends on the existing color of the walls in the room, and the aesthetic you're going for.
As a note, the new lighting temp will probably be jarring at first as it's much colder than "normal" warm tungsten screw-in light bulbs. I would consider the LED uplighting as an accent to your existing lighting, so you can turn it off or dim the LED system so it won't be overpowering in the evening. Also, there are some LED systems now that are color-programmable and even controlled by a mobile app, so you can pick the white color temperature that appeals the most to you, or have it cycle though different color temps throughout the day, light real sunlight.
Experiment with some simple uplighting cans on the floor and in some uplighting and work from there... if you are thinking of spending "real" money, ask an interior designer or decorator for advice.