Let $N_t$ be a numeraire and $(W_t)$ be the standard Brownian motion under the risk-neutral probability measure $P$.
Recall that forward measure $\hat{P}$ is defined as the Radon-Nikodym derivative: $$\frac{d\hat{P}}{d P} = e^{-\int_0^t r_s \,ds}\frac{N_t}{N_0}$$ where $r_s$ is risk-free interest rate.
Whenever I want to change the underlying measure to forward measure (take bond as numeraire), I always uses the equation $$d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t.$$ However, I am not able to prove that equation above implies that $(\hat{W}_t)$ is a Brownian motion under the forward measure $\hat{P}$.