The answer is: Yes, you can! Malay and Indonesian are very similar.
The Malay language (bahasa Melayu) has a large number of local dialects. Some of them are spoken in Malaysia, such as Johor or Kelantan Malay. Others are spoken in Indonesia, including Jambi, Minangkabau, Jakarta, or Manado Malay. Besides these regional dialects, the Malay language also consists of two major branches, namely Indonesian-Malay known as bahasa Indonesia, and Malaysian-Malay known as bahasa Malaysia. The meaning of bahasa is ‘language’, and in English we call he two languages Indonesian and Malaysian (or Malay). These two languages server as the naional language of four countries. Malaysian Malay is the national language in Singapore and Brunei Darussalam. Indonesian is the national language of the Republic of Indonesia.
There are certain differences between formal bahasa Malaysia and formal bahasa Indonesia, but the languages are very similar. Reading a text, or listening to a speech delivered in formal language, it can take a while before one knows whether the language is Indonesian or Malaysian.
In colloquial speech, the differences are naturally much more pronounced. But even between different regions within Indonesia and also within Malaysia there are considerable differences. Jakarta and Kupang Malay, noth spoke in Indoesia, are very different from each other. between, let’s say, Jakarta and Kupang Malay (both spoken in Indonesia). So are Kelantan and Johor Malay, spoken in Malaysia.
Formal Indonesian and formal Malaysian are thus very similar, and mutually intelligible (see our chapter on the history of Malay and Indonesian).
The Language used in «The Indonesian Way»
The language that we teach in «The Indonesian Way» is bahasa Indonesia. There are some differences between the two languages, but when you speak bahasa Indonesia instead of bahasa Malaysia in Malaysia, people can still understand you. Every educated Malaysian is aware of the subtle differences between Malaysian and Indonesian, and will not have any difficulties understanding you even if you speak with a distinct Indonesian accent! This is partly due to the popularity of Indonesian films and songs in Malaysia.
If you want to learn more about the history of the Malay language and the relation between Indonesian and Malaysian, read J. Collin (1998) “Malay, world language: a short history” or the Wikipedia article “History of the Malay language“