Store tomatoes with their stems down
Storing tomatoes upside down apparently makes a huge difference in how long they keep before spoiling. The area where the stem connects to the fruit is called the scar of the tomato; when the stem has been removed, air gets in to the tomato through the scar, moisture leaves the tomatoes through the scar, and all of that causes the tomato to spoil faster. If the stems are no longer attached, store your tomatoes upside down and they should last longer.