SVM radial basis generate equation for hyperplane
I'm not sure if I've fully understood you. Radial basis kernel assumes that you transform your features into an infinite space and the dot product of your transformed vectors is exactly the radial basis kernel.

$k(x,y)=\phi(x)\cdot \phi(y)$

$\phi(x)$ - mapping

The main reason for using a kernel trick is the ability to transform features into higher dimensions without knowing the map function explicitly. Your hyperplane has infinite number of coefficients. You can always expend the radial basis kernel into Taylor series and get some of the initial coefficients.