If you look at the Galaxy map, when you select or hover the cursor over a star, a little popup flag appears with the star name, some basic details, and several icons. The icon that looks like a little sun with planets orbiting it is the System Map button. This button is red for Unexplored systems and white for Explored systems. Since there are no systems for which the default status is Explored beyond 500 LY from Sol, every star out there with a white System Map button has been explored by you.
Sorry, but this is the only in-game tool to allow you to see this. An option to show Explored/Unexplored stars more clearly on the starmap was discussed back in the development stage as a desired feature, but is yet to be implemented.