I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where: $$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$ $$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$
We know that $W_1$ and $W_2$ are Brownian motions. Generally, I tried to use a multidimensional version of Ito's lemma. I used this formula:
$$dV = \Bigg(\frac{\partial V}{\partial t} + \frac12 \sum_{i=1}^d \sum_{j=1}^d \sigma_i \sigma_j \rho_{ij} S_i S_j \frac{\partial^2 V}{\partial S_i \partial S_j}\Bigg)dt + \sum_{i=1}^d \frac{\partial V}{\partial S_i} d S_i.$$
I got:
$$dV = \Bigg( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma_1^2 S_1^2 \frac{\partial^2 V}{\partial S_1^2} + \sigma_1 \sigma_2 S_1 S_2 \frac{\partial^2 V}{\partial S_1 \partial S_2} + \frac12 \sigma_2^2 S_2^2 \frac{\partial^2 V}{\partial S_2^2} \Bigg)dt + \frac{\partial V}{\partial S_1} dS_1 + \frac{\partial V}{\partial S_2} dS_2.$$
Is it a proper way to determine the Black-Scholes formula for this option? What should I do next?
