We present a general method to obtain a closed finite formula for the exponential map from the Lie algebra to the Lie group for the defining representation of orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group and is also independent of the metric. We present an explicit formula for the exponential of generators of the SO₊(p,q) groups with p+q=6, in particular, dealing with the conformal group SO₊(2,4) which is homomorphic to the SU(2,2) group. This result is needed in the generalization of U(1)-gauge transformations to spin-gauge transformations where the exponential plays an essential role. We also present some new expressions for the coefficients of the secular equation of a matrix.