One general construction can be found in Revuz and Yor "Continuous Martingales and Brownian Motion" for instance:
proposition (1.3)
Let $H$ be a separable real Hilbert space. There exist a probability space $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ and a family $X(h)$, $h \in H$ of random variables on the space such that
i) the map $h \rightarrow X(h)$ is linear
ii) for each $h$ the random variable $X(h)$ is Gaussian centered and $\mathbb{E} [ X(h)^2] = ||h||_{H}^{2}$
Nualarts book "Malliavian Calculus" also starts with the notion of isonormal Gaussian process which is general as well as Adler's books on Gaussian processes.
Alternatively you could look at one of T. Hida's books on White noise analysis for a construction based on the bochner-milnos theorem and Nuclear spaces.
Sorry none of these have a geometric perspective that I am aware of...