Birkhoff-Sternberg normal form coordinates around a hyperbolic fixed point allows one to (almost) linearize the dynamics locally. When working with flows, a common approach is to study the Poincare first return map and return times independently. We provide a new approach which finds coordinates putting the generating vector field of a flow in a normal form on a tubular neighborhood of a periodic orbit. Furthermore, when the flow is a Reeb flow of a contact form, we put the pullback of the contact form in a normal form as well. Joint with A. Erchenko and Y. Yang.