<p>Self-orthogonal codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs, lattices and linear complementary dual (LCD for short) codes. One of the major approaches of constructing self-orthogonal codes is the employment of some special functions, and it is in general hard to design self-orthogonal codes with good parameters and determine their parameters. In this paper, we investigate this approach further by using square functions over finite fields, thereby obtain some ternary self-orthogonal codes and explicitly determine their parameters. The parameters of these self-orthogonal codes are new and flexible. Furthermore, the results of this paper show that these obtained self-orthogonal codes have a few nonzero weights with at most five and some of these codes are minimal.</p>