In the present paper the authors constructed a \(\mathbb{Z}_{2}\)-valued spectral flow for paths of skew-adjoint Fredholms on a real Hilbert space. First the authors defined the \(\mathbb{Z}_{2}\)-valued spectral flow associated to a straight line path in finite dimensions. The definition simply counts the number of orientation changes of the eigenfunctions at eigenvalue crossings through 0 along the path. Then the authors gave the analytic approach to the complex spectral flow for paths of self-adjoint Fredholm operators on a complex Hilbert space, this allows to show relatively directly that the \(\mathbb{Z}_{2}\)-valued spectral flow can be calculated, similarly to the complex spectral flow, as a sum of index type contributions, provided the appropriate notion of index is used. Finally an index formula is proved which connects the \(\mathbb{Z}_{2}\)-valued spectral flow of certain paths in the skew-adjoint operators on a real Hilbert space to the \(\mathbb{Z}_{2}\)-index of an associated Toeplitz operator on the complexification. At the end the authors illustrated the theory by an explicit example given by a matrix-valued shift operator which can be considered to be the analogue in real Hilbert space of the standard Toeplitz operator in the complex case. This example is the canonical non-trivial example of \(\mathbb{Z}_{2}\)-valued spectral flow.