The classical Bernoulli polynomials \(B_n(x)\), \((n=0,1,2,\cdots)\) may be defined in several different ways as summarized in the beginning of the paper under review, such as using the generating function given by Euler, or as an Appell sequence. As the main result of the paper, a new approach for Bernoulli polynomials is proposed. It is shown that \(B_n(x)\) can be expressed as the determinant of an upper Heisenberg matrix of order \(n+1\) \[ B_n(x)=\frac{(-1)^n}{(n-1)!}\left| \begin{matrix} \quad& \quad& \quad& \quad& \quad& \quad & \\ 1&x&x^2&x^3&\cdots&x^{n-1}&x^n\\ 1&\frac12&\frac13&\frac14&\cdots&\frac{1}{n}&\frac{1}{n+1}\\ 0&1&1&1&\cdots&1&1\\ 0&0&2&3&\cdots&n-1&n\\ 0&0&0&\binom{3}{2}&\cdots&\binom{n-1}{2}&\binom{n}{2}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&\cdots&\binom{n-1}{n-2}&\binom{n}{n-2}\\ \end{matrix} \right| \] by proving its equivalence to either definition stated above. Some well-known properties of Bernoulli polynomials are recovered using basic properties of determinants.