Yes, tomato paste should generally be cooked uncovered to allow excess moisture to evaporate and to concentrate its flavor. Cooking it uncovered also helps to prevent splattering.
Cooking tomato paste is a crucial step in many recipes because it intensifies its flavor and removes the raw, acidic taste. When you cook tomato paste uncovered, the water content gradually evaporates, leading to a richer, more concentrated tomato flavor. This process also allows the sugars in the tomato paste to caramelize, adding depth and complexity to the dish. The heat helps to break down the paste, making it easier to incorporate into the other ingredients. If you were to cook it covered, the steam would trap the moisture, hindering the caramelization and flavor concentration process. Start by adding the tomato paste to a pan with a little oil or butter over medium heat. Stir frequently to prevent burning, and cook until the paste darkens slightly and develops a fragrant aroma. This usually takes about 5-10 minutes.
To prevent tomato paste from sticking and burning, especially in stainless steel pans, add a pinch of salt to the pan along with the oil or butter. The salt helps to draw out moisture and prevents the paste from scorching.