We discuss a generalization of geometric algebras known as ternary Clifford algebras. In these objects, we have a fixed ternary form instead of a quadratic form as in ordinary geometric algebras. Basis-free definitions of the determinant, trace, and characteristic polynomial in ternary Clifford algebra are introduced. Explicit formulas are presented for all coefficients of the characteristic polynomial and inverse in ternary Clifford algebra. The operation of Hermitian transpose (Hermitian conjugation) in ternary Clifford algebra is introduced without using the corresponding matrix representation. We present a natural realization of the unitary Lie group \(\textrm{SU}(3)\) , which is important for physical applications, using only operations in ternary Clifford algebra. An explicit basis of the corresponding Lie algebra \(\mathfrak {su}(3)\) is presented. We present an explicit connection with the well-known Gell-Mann basis of \(\mathfrak {su}(3)\) . The results can be used in physics, computer science, and engineering.