Algorithms can't work over $\mathbb{R}^n$, as you can't represent real numbers in finite space.  Also, you can't pick a number uniformly at random from $\mathbb{R}$.  Instead, usually we work over a finite field.

Then we can't do any better.  Suppose we are working in the finite field with two elements, $GF(2)$.  Suppose that

$$AB - C = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.$$

Then it is easy to verify that Frievald's algorithm is wrong with probability $1/2$, as $(AB-C)v = 0$ holds with probability $1/2$ when we select $v$ uniformly at random.  You can generalize this to a $n\times n$ matrix that is all zeros except for a single entry, and then the probability of false positive is $1/2$.