One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product.  So what you're asking for is a special case of extending the notion of standard Gaussian measure to a particular infinite-dimensional space.  But [you can't do that.][1]


  [1]: https://mathoverflow.net/a/36417/1044